• To demonstrate an understanding of limits. Supported by learning objectives 1, 2, and 3.
• To demonstrate an understanding of continuity. Supported by learning objective 4.
| C.1 |
| C.2 |
| C.3 |
| C.4 |
Objectives
C.1 Back to Top
To gain an understanding and an appreciation for the meaning of limit.
To engage in activities that require exploration and manipulation to develop understandings of the concept of limits (CCT).
Instructional Notes
The software, Journey Through Calculus (see Calculus 30, A Bibliography), has a very interesting demonstration of a weightlifter that could be used to introduce the students to limits.
The purpose of this section is to give the students a hands-on physical experience with limits. These activities can be used as group projects or personal investigations. Note that most of these activities are examples of one-sided limits.
1. Newspaper Cut: How many times can you fold a sheet of newspaper and still cut through the folded newspaper with a scissors?
2. Duct Tape: What is the minimum number of pieces of duct tape of a given length needed to suspend a certain object (brick, student, other convenient mass) against the wall above the floor?
3. Balloon Pop: How many full puffs of air does it take to inflate a balloon to the bursting point? Alternately, what is the maximum circumference a spherical balloon can reach before it pops?
4. Elastic Band: What is the maximum stretch of a particular size and type of elastic band? This can be determined by securing one end of the elastic band and adding weights to the other until it snaps.
5. Humpty Dumpty: What is the maximum mass a raw egg, seated vertically, can hold before it breaks? Take an empty 2 L carton of milk and cut out viewing windows at the bottom of the carton so that you'll be able to see the egg. Place the egg at the bottom of the milk carton seated inside a portion of an egg carton to hold the egg in a vertical position. Use an empty 1 L milk carton to act as a plunger inside the 2 L carton. It is not a perfect fit, but if the egg is wedged into a corner of the bottom of the 2 L carton, it should work. Begin to fill the 1 L carton with water as it rests on top of the egg. Continue adding water, little by little, until the egg breaks. To measure the mass of the water, pour it into a graduated cylinder to find its volume. Since 1 ml of water has a mass of 1 g, you can determine the maximum mass an egg can hold. Sand could be substituted for water. A balance scale would be needed to determine the mass of the sand (CCT).
Examples/Activities
A program that you could run on your TI-83 calculator to simulate the weightlifter demonstration from the Journey Through Calculus software appears below.
ClrHomeSuggestions/Extensions
As students carry out activities, such as those suggested, have students discuss what has happened mathematically to capture the idea that the situation has an upper or lower limit. The weightlifter activity provides for the students to get closer and closer to the limit value without ever reaching the limit. Students can compare this to other activities to discuss how, in some cases, the limit can be attained and, in other cases, the limit can only be approached. The weightlifter activity can also be adapted for a student or teacher to role play the weightlifting.
Have the students also reflect on the limits (attainable and approachable) that they have seen in their graphing of functions. For example, piecewise functions and functions with asymptotes can provide lots of opportunities for defining and discussing limits. Students can also be asked to consider how the word “limit” is used in everyday language and relate those situations to their development mathematical understanding of limits.
Objectives
C.2 Back to Top
To use limit notation, to evaluate limits, and to determine if a function is continuous by examining the graph of the function.
To justify decisions based upon understanding of calculus concepts (CCT).
Instructional Notes
Using limit notation and knowing if a function is continuous are two skills that are essential to understanding differentiation of functions.
Ask students where, in everyday life, they have used or heard others use the word “limit”. The discussion will likely move to the notion of a boundary as suggested by speed limits and load limits on highways. Not all limits are as easily determined. The more mass that is attached to an elastic band, the longer the band becomes. Suppose you know that an elastic band will snap if 300 g or more is attached to it. Ask the class how they would determine the maximum length that the elastic band reaches before it snaps? As the attached mass, m , approaches 300 g (from values smaller than 300), the length of the band, L , approaches its maximum length B . This discussion could lead nicely to the notation 
(approaches) and you could write “If 
and 
, 
“. Have the students actually try an experiment similar to this one. You will not, however, know the maximum mass that the band can support as in the fictional example.
Students could also experiment with the formula 
familiar to them from Math B30. This formula gives the amount, A , that an investment of P dollars will become if invested at an annual interest rate of r % for t years if interest is compounded n times per year. Fix the interest rate at 12%, the time period for 1 year, and the amount invested at $10 000. Then the formula becomes 
. Set up a table and use a calculator to determine what happens to the $10 000 investment as you increase the number of times the interest is calculated. Use this example to introduce the notation 
. As 
, 
to the nearest cent. This is an excellent example to show the limitation of a calculator in determining the limit of a function. Calculators differ, but if you make n too large, your calculator may give results that are inappropriate.
Ask students how they could determine an approximation to the circumference of the unit circle if they did not know the formula 
. Recall the law of cosines: 
. For an isosceles triangle with legs 
and vertex angle 
, we have 
. Thus 
.
Suppose a regular octagon is inscribed in the circle. Then 
, and the perimeter of the octagon is 
. We can set up an expression for the perimeter of an n -sided regular inscribed polygon to be 
. As the number of sides increases, the perimeter of the n -gon approaches the circumference of the circle. Students could use their calculators to approximate the limit as n approaches 
of the expression 
. You might introduce the notation 
.
When students have experienced the use of the word “limit” in a practical setting, as in the last three examples, move to finding limits by examining a function and its graph. Ask the students to determine what happens to the output values of the function 
as x gets closer and closer to 3. Complete a table such as the following.
By examining the table, we see that as the input values approach 3 from the left, the output values approach 6. Similarly, as the input values approach 3 from the right, the output values approach 6. Graphically we see the same situation.
This leads us to say that as x approaches 3, 
approaches 6. Using arrow notation, we can write as 
, 
. The number 6 is said to be the limit of 
as x approaches 3. We abbreviate this sentence by writing 
or 
replacing 
by the expression for the function 
which we read as “the limit of 
, as x approaches 3, is 6”.
Establish the following definition .
A function f has the limit L as x approaches a , written 
, if we can get 
as close to L as we wish by restricting x to a sufficiently small interval about a , but excluding a . ( Calculus , Marvin Bittinger, 4 th edition).
The last part of the definition “but excluding a ” is very significant. This tells us that the value of a limit as x approaches a does not depend on the function's value at a . It may very well be that the function is not defined at a . What matters is how the function behaves near a . See the following graphs for examples.
In each of the following three cases, 
even though in figure (ii) 
and in figure (iii) 
is not defined. The open circle indicates the function does not have a value for 
.
Next introduce situations in which the limit of a function does not exist. Limits may not exist for several reasons. Consider first the function 
whose graph is shown below.
As input values approach 2 from the right, output values become more and more positive without bound, whereas as input values approach 2 from the left, output values become more and more negative without bound. We say that 
does not exist.
In order for the limit to exist at 
, the limit as you approach a from the left and the limit as you approach a from the right must exist and be the same real number .
Consider the graph of the piecewise function 
shown below.
As input values approach 0 from the right (denoted by 
), output values approach 1. As input values approach 0 from the left (denoted by 
), output values approach 
. As the limit from the right, 1, is not the same as the limit from the left, 
, the limit does not exist and we write 
does not exist.
Consider the graph of the function 
shown at right. As input values approach 0 from either side , output values become increasingly more negative without bound, and we write 
. The symbol 
does not represent a real number. The limit does not exist. By writing 
, we communicate more clearly the manner in which the limit does not exist.
Further examination of the function reveals that 
while 
. Thus 
does not exist.
It is critical to get the terminology correct. A limit exists at 
if approaching 
from either side, the function approaches the same real number . Otherwise, a limit does not exist. If a limit does not exist, do not say the limit is undefined. Rather, if the output values as you approach 
from either side are become increasing large positive (or negative), indicate that the limit is 
( 
). If the output values as you approach a from one side are increasingly large positive, but as you approach from the other side the output values are increasing large negative (or vice versa), then say the limit does not exist (abbreviated d.n.e.).
By examining the graph of 
above, we may conclude that 
and 
assuming the x -axis is a horizontal asymptote line. The symbols 
( 
) are used to indicate that the input values are approaching a large positive (negative) number.
In random order, show the graphs of several functions that are continuous and those that are not. Use a concept attainment activity to have the students determine the property that is common to the two categories (CCT). Informally, a function is continuous over , or on , some interval of the real number line if you can trace the graph of the function in that interval without lifting your pencil from the page . Thus continuous functions will not have jumps, rips (vertical asymptotes), or holes (CCT).
The functions below cannot be drawn without lifting your pencil. Each illustrates a different reason for being discontinuous.
You cannot trace 
without lifting your pencil because there is a hole in the graph at 
. The function is not defined for 
. You cannot trace the graph of 
without lifting your pencil, because as you approach c from the left, the function's values approach t , but as you approach c from the right, the function's values approach s . There is a jump in the function. You cannot trace the graph of 
without lifting your pencil, because as you approach c from the left or right, the function's values approach t , but at 
, the function's value is s .
If a function's value at 
is the same as the limit as x approaches c from either side, the function is said to have continuity at the point 
. In the figure at right, the function 
is continuous at the point 
, because 
.
Students must know and be able to test for all three parts to the following definition.
A function 
is continuous at the point 
if:
(a) 
exists (it is some real number)
(b) 
exists (remember to approach a from both sides)
(c) 
(the function's value must equal the limit value).
When the students understand continuity by the pencil test, expect the students to explain why a function is continuous (not continuous) with respect to the three conditions for continuity.
Discuss the continuity of each graph shown in this section. For functions that are not continuous, indicate which of the three parts of the continuity definition are not satisfied by the function.
In view of objective C.1, it is important to discuss, without proof, the following continuity principles .
Continuity Principles1. All constant functions are continuous.
2. The following types of functions are continuous at every member in their domain: polynomial, rational, power, root, trigonometric, exponential, and logarithmic.
3. If 
and 
are continuous functions, so are 
, 
, 
, and 
provided, in the last case, that 
.
We can use these principles to explain why 
is continuous. By principle 2, 
and 
are continuous functions because they are polynomial functions. By principle 3, 
must be continuous because 
is never 0 for real values of x .
Examples/Activities
1. Students could explore the expression 
for different values of k .
(a) What happens to the value of the expression as k gets close to 0?
(b) What happens to the value of the expression, as k gets very large?
(c) How large does k have to be before the value of the expression does not change in the first three decimal places?
(d) Why isn't the value of the expression getting close to 1? After all, as k gets large,
, you would expect the expression to approach
or 1.
Introduce the students to the name of the number, 
, that is being approached and then write the limit expression for e .
2. Explore the expression 
for increasingly large values of n .
How does it compare to the value of the expression in 1(b)? Write a limit expression.
3. Find the sum of two, three, four, five, . . . terms in the series below.
Newton used this series to calculate 
to 14 decimal places. The sum of 30 terms in this series will give 
to 19 decimal places. Write a limit expression for 
.
4. Are 
and 1 different? Explain (CCT).
5. Working in groups of 3 or 4, have the students explore the relationship between the area of a circle and the area of a parallelogram constructed from the circle (divide the circle into equal-sized sectors and rearrange the sectors into a parallelogram-like form). Have the students discuss the approximate height and length of the parallelogram in relation to the radius of the circle and to use those expressions to approximate the area of the circle (students may assume the circumference of the circle to be 
). Have the students repeat this with circles cut into more wedges and discuss how the shape is getting closer to an actual parallelogram, and therefore, their estimation of the area, 
, is becoming more accurate. It is important that students be encouraged to begin to use some of the appropriate terminology. For example, the students could talk about “as the number of wedges increases” or “as the number of wedges gets closer to infinity” or “as the number of wedges approaches infinity”. Other terms that should be introduced in context are “limit” and “converges”.
6. Have the students research Zeno's (or Xeno's) Paradox and then have them explain why the paradox is related to limits.
1. Use your calculator to complete a table similar to the one below. From your table, estimate 
for each of the following functions. (NUM)
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(work in radians)
Students should carry out a series of exercises similar to the ones below to determine if a function is continuous over an interval and also to determine the value of a limit visually by examining the graph of a function. Exercises like these can be found in most recent Calculus texts. (Note: Continuity is described later in this section.)
1. By examining the graph of the function below, answer the following questions.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
(k) 
(l) 
2. Is the function 
above continuous on each of the following intervals? Use the “pencil trace” test.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
Students should have an opportunity to draw graphs of functions that are not continuous due to different conditions.
1. Sketch the graph of a function that is not continuous at 
because:
(a) the function is not defined at 
, however 
exists.
(b) the function is not defined at 
, and 
does not exist.
(c) 
, but 
is defined.
(d) 
even though 
and 
exist.
Determine if the function 
, whose graph is shown below, is continuous at the point with the given x -coordinate – parts (a) to (g). If you determine that the function is not continuous, explain on which continuity condition(s) it has failed (CCT). Be very specific. For example if the function is not continuous at 
, you would respond with one of the following three statements:
• “The function is not continuous at 
because 
does not exist.”
• “The function is not continuous at 
because 
does not exist.”
• “The function is not continuous at 
because 
.”
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
1. Provide an argument, using the three continuity principles, as to why each of the following functions must be continuous (NUM).
(a) 
(b) 
(c) 
(d) 
(e) 
Suggestions/Extensions
Some limits that students might like to consider and discuss include how far you can hit a golf ball, what speed you can reach after travelling 400 m, how long a light bulb will burn, the number of people our earth can support.
Write a computer program to find the sum of the series 
. Find the sum for “large” values of n (like 1000,
10 000). What number does the sum of the series seem to approach? What appears to be the result of 
?
Ask the students to try and determine an approximation to the area above the x- axis but under the curve 
. Have them suggest ways to divide up the region using shapes whose areas are easy to determine. For example, rectangles or trapezoids. The focus of this activity is the development of the concept of a limit, not integration.
You can make efficient use of a TI-83 calculator to complete a table such as the one at left by proceeding as follows.
(a) enter the function under Y=
(b) On the home screen key in 
and press ENTER.
(c) Press 2nd ENTER to return the function.
(d) Use your arrow keys and the Insert feature to change the x value.
(e) Press ENTER to find the new y value.
Ask the students to sketch a graph that shows different possibilities for 
and then write the corresponding limit statements.
Have the students develop the three conditions for continuity based on their experiences with the pencil test for continuity.
Have the students discuss why the second continuity principle contains the phrase “at every member in their domain”.
Objectives
C.3 Back to Top
To determine limits, at real numbers and at infinity, of many types of functions including those that involve absolute value, roots, and piecewise functions, by applying appropriate strategies such as direct substitution, factoring, simplifying, rationalizing, and rewriting.
Instructional Notes
This objective differs from C.1 in that we are not determining limits from the graph of a function nor are we guessing as to what the limit might be by examining a table of values created with the assistance of a calculator. The emphasis, here, is for students to determine the limit, step by step, through a sequence of connected steps, or by creating a sign analysis of the function.
Emphasize that each step must be connected to the next using an equality sign and must include the entire expression whose limit is being evaluated (in parentheses, if the expression consists of more than one term). Thus if we wish to evaluate the limit of the expression 
as 
, we write 
and not 
. This is similar to logarithmic functions or trigonometric functions: 
is not always the same as 
, 
is not always equal to 
.
Ask students to find a limit such as 
. From their work in C.1 students would know that 
is continuous. Thus, the limit can be evaluated by direct substitution.
.
Begin each limit question by trying direct substitution. If direct substitution of 
leads to the form 
where 
, then the function has a vertical asymptote with equation 
(Math B30). You will need to perform a sign analysis on the function to determine if the limit does not exist (the function goes to 
on one side of the asymptote but to 
on the other side) or if the limit can be better described as being either just 
(the function goes to 
on both sides) or 
(the function goes to 
on both sides).
Have students try to determine 
. Direct substitution yields 
so a vertical asymptote line occurs at 
. Perform a sign analysis of 
to see how the function behaves near 2.
From the sign analysis, we see that 
while 
. Thus, 
does not exist.
Determine 
. Direct substitution yields 
so we will use a sign analysis of 
. From the sign analysis we can see that 
. Similarly, 
. Thus, we conclude 
. Note that we do not say that the limit does not exist since 
gives a clearer picture of the way in which the limit does not exist.
If direct substitution yields 
, you may have to factor, simplify, or rewrite the expression in a different form in order to evaluate the limit.
Factoring
Ask the students to determine 
. Direct substitution yields 
, which is indeterminate. As we look at the expression, we see that the numerator is factorable. We can write 
. Because 
, and is only approaching 2, we can reduce the expression and write 
. The graph of 
is simply that of 
provided 
. As 
, 
. As the function is indeterminate at 
, the graph of the function has a hole at 
.
Hole Location Principle
If the function 
yields 
for 
, and if 
, then the graph of the function has a hole at the point with coordinates 
.
Simplifying
Ask the students to try and determine 
. Direct substitution yields 
. The expression does not look factorable. The numerator, however, can be simplified: 
.
There are three things to watch for. First, students have trouble knowing when to quit writing the 
. This is dropped the moment you substitute. Second, students inadvertently place the 
sign between the 
and the expression rather than in front of 
. Finally, having students work down the page, not across as has been done here, makes it easier for you to follow their work.
Note: the first two problems indicate that the students are still not clear as to what a limit is and what the notation stands for, so some review should be undertaken.
RationalizingSome limits can be determined by rationalizing the numerator (or denominator). Have the students consider 
. Direct substitution yields 
, and it appears the expression can be neither factored nor simplified. By eliminating the radical from the numerator, the limit can be determined:
. Have the students discuss why it is important to not multiply out the expression in the denominator (in this case).
Limits of Rational Expressions at Infinity
When looking at limits as 
or 
, it is important for the students to know that they cannot use direct subtraction because 
and 
are not numbers.
Begin with a discussion as to what happens to expressions of the form 
, 
, 
, 
, 
, 
as 
or 
. Through this discussion, informally establish the results that 
, and 
where a is a real number and n is a positive integer. Follow this up with a discussion as to what happens to 
, 
, and 
as 
. As we can make these expressions as large as we like by taking larger and larger values of x , we have 
, 
, and 
. Through informal discussion, establish 
, 
, and 
.
Have students consider a limit such as 
. Students who remember their Math B30 course should recognize this as a rational function with the degree of the numerator less than that of the denominator. They may recall that this causes the function to be asymptotic to the x -axis and thus as 
, the function approaches 0. The purpose of this section, however, is to reach that same conclusion through a series of connected steps involving limits.
In problems involving limits of rational functions at infinity, it is usually best to divide both numerator and denominator by the highest power of x in the denominator.
Thus, 
. Note that based on the discussion, students should realize that 
, 
, and 
all approach 0 as 
.
Do not let students substitute 
in place of x . They should not be writing 
.
Ask the students to determine 
. Dividing by the greatest power of x in the denominator, we have 
. The last step is a bit of a jump. Both x and 
become arbitrarily large as 
, while the denominator approaches 3. Thus, the quotient grows towards 
. To evaluate 
, write 
. Then reasoning that both x and 
become arbitrarily large, so will the product. Thus, 
. Visualizing the graph of 
leads to the same conclusion.
Limits of Radical Expressions at Infinity
Recall, from A.5, that 
, whereas 
. Ask the students to try to evaluate 
. By removing a factor of 
from the expression beneath the radical sign, we have 
. Since 
, we know that 
because 
. Thus, we have 
. Next, have students determine 
. The key step is to realize that if 
. Then 
must be replaced by 
and not x . The value of the limit becomes 
.
It is easiest to evaluate limits involving absolute value if you first remove the absolute value sign from the expression. You will have to know whether to replace 
with 
or 
. Have students consider 
. Direct substitution of 
yields 
: 
. Note the use of the absolute value property 
from A.5. If 
, then the values of x are smaller than 4. Thus, 
would be negative, so we replace 
by 
. Thus, 
. Next, determine that 
(note this time 
is replaced by 
). As 
, then 
does not exist. Some students may choose to use a number line test to determine the sign to use on the absolute value limits.
Limits Involving Split Functions
Ask students to determine 
if 
. Note that the split in the function occurs at 
. Thus to determine 
, we must first consider both 
and 
. When examining 
, we consider that part of the function for which 
, namely 
. Thus, 
. To find 
, we must consider that part of the function for which 
, namely 
. Thus 
. As 
, then 
does not exist.
Examples/Activities
Be sure to include in your assignment, questions that lead to the form 
, 
as well as limits of constant functions. Both can cause confusion for students as they are trying to sort out the meaning of limits and functions.
1. 
2. 
3. 
4. 
5. 
Evaluate each of the following limits by performing a sign analysis on the function.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
Evaluate each of the following limits by factoring first.
1. 
2. 
3. 
(yes, you can factor 
--objective A.2)
4. 
5. 
(factor 
as a difference of cubes)
6. 
(do not rationalize, factor)
7. 
(you can factor 
– see objective A.2)
Explain why it is not completely correct to say 
, but it is correct to say that 
.
Determine the coordinates of any holes in the graphs of the functions below.
1. 
2. 
3. 
(there are two)
4. 
5. 
6. 
(there are two)
Examples/Activities
Evaluate each of the following limits. Begin by simplifying. You may then have to factor.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
Determine each of the following limits by rationalizing.
1. 
(do not factor)
2. 
3. 
4. 
(do not factor)
5. 
6. 
Hint: 
. Applying that here, to rationalize the denominator, multiply numerator and denominator by 
.
Examples/Activities
Evaluate each of the following limits.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
Examples/Activities
If 
, then find each of the following limits.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) were (d) and (e) necessary to determine (f)? Explain.
If 
find each of the following limits.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
Is 
given above continuous at
(a) 
?
(b) 
?
After students have practised evaluating limits from each of the categories, it is essential that students be asked to evaluate several limits from the different categories (with the questions coming in random order). Often students have difficulty recognizing the appropriate strategy to apply.
Suggestions/Extensions
If a rectangle of width w is inscribed in a circle of radius 10 cm, then the ratio of the area of the rectangle to the area of the circle is given by the expression 
.
(a) Find 
and explain why the answer makes sense.
(b) Find 
and explain why the answer makes sense.
A rectangle has vertices with coordinates 
and 
. If the midpoints of all sides of the rectangle are connected in order, a rhombus is formed. Find 
.
A company estimates that the cost in thousands of dollars to remove x % of pollutants created by its manufacturing process is given by 
.
(a) Find the cost for the company to remove 20% of pollutants.
(b) Find the cost for the company to remove 60% of pollutants.
(c) Find the cost for the company to remove 90% of pollutants.
(d) Find 
and interpret the result.
A farmer's square house has dimensions a metres by a metres. The house is placed in the corner of a square farmyard of dimensions x metres by x metres. Region I is a lawn area and is cut and watered weekly. Region II is a garden area and is watered weekly.
(a) Find an expression in terms of x and a for the area of the garden.
(b) Find an expression in terms of x and a for the area of the lawn.
(c) Find an expression in terms of x and a for the ratio of the area watered to the area cut.
(d) Find 
of the expression in part (c). What does this physically represent?
(e) Find 
of the expression in part (c). What does this physically represent?
The dollar cost to courier a package of mass m kg is given by the formula 
.
(a) Find the cost to courier a package whose mass is 875 g. Be sure to note the units.
(b) Find the cost to courier a package whose mass is 7.44 kg.
(c) Find 
.
(d) Find 
.
(e) Explain why you cannot find 
.
Can You Be As Old As Your Mother?
Suppose you were born on your mother's birthday when she was 24 years old.
(a) What is the ratio of your mother's age to your age on your first birthday?
(b) What is the ratio of your mother's age to your age on your 12 th birthday?
(c) What is the ratio of your mother's age to your age on your x th birthday?
(d) Find the limit as 
of the expression in part (c) and explain what the result means.
You might ask students to examine, using a table of values, 
and 
if a is a real number and n is a positive number (not necessarily a positive integer).
Ask the students to justify the statement that “in problems involving limits of rational functions at infinity, it is usually best to divide both numerator and denominator by the highest power of x in the denominator”.
What does doing this action cause and why is the result beneficial? Encourage the students to explain what is happening in limits approaching infinity by talking about factors that will become arbitrarily larger or smaller and what that means. Some students will prefer to talk about the degree term of the numerator and denominator and how the two relate to each other. For these students, they should be able to explain why those terms can be the sole focus of their evaluation of the limit.
Suppose you travel from Saskatoon to Regina at an average speed of x km/h. On your return trip, suppose you average y km/h. If the average speed for your entire trip is 80 km/h, then x and y are related by the equation 
.
(a) Find y if 
km/h.
(b) Find y if 
km/h.
(c) Find y if 
km/h.
(d) Find 
.
(e) Explain what this limit tells you.
A company estimates that its computer equipment depreciates $1000 the first year, $800 the second year, $640 the third year, and so on each year depreciating 80% of the amount it depreciated the year before. After n years, the total of all the depreciations is 
.
(a) What is the total of all depreciations after 5 years?
(b) Find 
. (Hint: what happens to the value of 
as 
?)
(c) What does this result tell you?
Objectives
C.4 Back to Top
To identify the location of a discontinuity and the type of discontinuity by examining the graph of a function or just the function.
To engage in activities that require exploration in order to develop understandings of the concept of discontinuities (CCT).
Instructional Notes
Being able to categorize discontinuities, using the appropriate vocabulary, aids in clear communication regarding a function's behaviour.
Discontinuities are often classified based on what the graph of the function looks like at the point of discontinuity. Show students graphs of functions having different discontinuities and have students suggest names for the graphs (CCT). Students may be able to come up with some of the actual names that are used.
Infinite Discontinuities
This name is used to describe discontinuities that occur at vertical asymptotes.
The function 
has two infinite discontinuities, one at 
and the other at 
(as shown at right).
Infinite discontinuities can be located by finding values of x that yield 
for 
where 
.
Step functions like the greatest integer function are good examples of functions that contain jump discontinuities. Piecewise defined functions and functions containing absolute values may also contain jump discontinuities.
The function 
contains a jump discontinuity at 
. Its graph appears above.
To find jump discontinuities of piecewise functions, pay particular attention to the behaviour of the function at the endpoints of the intervals at which the definition of the function changes.
To determine if a function containing an absolute value symbol has a jump discontinuity, pay particular attention to the behaviour of the function at values of x at which the expression inside the absolute value changes signs.
Removable Discontinuities
Introduce removable discontinuities by showing students graphs of functions containing removable discontinuities and graphs of functions containing nonremovable discontinuities, and ask students what, if anything, can be done so that the function will be continuous.
has a removable discontinuity at 
if 
can be defined or redefined at 
so that it can be made continuous. Speaking informally, a function has a removable discontinuity 
if you can remove or plug the hole in the graph by assigning an appropriate value to 
.
Summary of How to Identify Discontinuties
1. If
, then
is a vertical asymptote line and thus there is an infinite discontinuity at
.
2. If
, and if
, then there is a removable discontinuity (hole in the graph) at
. The coordinates of the hole are
. If
, yet both limits exist, then there is a jump discontinuity at
. Jump discontinuities tend to turn up in piecewise functions or in functions that contain absolute value symbols. For piecewise functions, be sure to examine the limit as x approaches (from both sides) the x values at which the definition of the function changes. Similarly, for functions containing absolute value signs, find the x values at which the expression within the absolute value symbols changes signs. Find the limit as x approaches these values (from both sides).
Examples/Activities
Give students some graphs of functions that are not continuous and ask them to identify the type of discontinuity. Students will find this relatively easy.
Next, give the students some functions and have them determine if the function is continuous or not. If it is not continuous, have students identify the value(s) of x at which any discontinuity occurs and classify the discontinuity. Some suggestions follow. (Note: Deal with all four types of discontinuities before using this set of exercises.)
a) 
b) 
c) 
d) 
e) 
f) 
g) 
h) 
i) 
j) 
k) 
l) 
m) 
n) 
o) 
p) 
q) 
r) 
s) 
Determine if the function is continuous or not. If it is not continuous, determine the value(s) of x at which any discontinuity occurs and classify the discontinuity.
a) 
b) 
c) 
d) 
e) 
Find the value of the constant p so that the function is continuous for all real numbers.
1. 
The solution for questions like this one should be written as shown.
Solution: 
and 
are continuous because all polynomial functions are continuous (Continuity Principles). For the split function to be continuous, we must be sure that at the value of x where the function is split, the y values are the same. In other words we must be sure that 
. Thus, 
. Solving the resulting equation 
, yields 
. Thus, for 
to be continuous, 
.
2. Find the value of the constants p and q so that the function 
is continuous for all real numbers.
Students' solutions should include 
and 
resulting in a pair of linear equations that can be solved for p and q .
Examples/Activities
Students should be able to define a function's value so that a discontinuity can be removed. (Hint: Use the hole location principle of C.3.)
(a) Define 
so that the function 
will be continuous at 
.
(b) Define 
so that the function 
will be continuous at 
.
(c) Define 
so that the function 
will be continuous at 
.
Students should be able to sketch the graph of a function given particular discontinuities.
(a) Sketch the graph of a function that is continuous everywhere except at 
where the function has an infinite discontinuity.
(b) Sketch the graph of a function that is continuous everywhere except at 
where the function has a jump discontinuity and at 
where the function has a removable discontinuity.
(c) Sketch the graph of a function that has an oscillating discontinuity at 
, a jump discontinuity at 
, an infinite discontinuity at 
, and a removable discontinuity at 
. It is continuous everywhere else.
Suggestions/Extensions
The function 
is said to be continuous from the right at 
, if 
. The function 
is said to be continuous from the left at 
, if 
.
The function below is continuous from the right at 
but not from the left. The function below is continuous from the left at 
but not from the right.
Note that graphing calculators do not usually reveal removable discontinuities as the location of the hole may not coincide with the points plotted by the calculator. By pressing TRACE on the calculator and using the arrow keys, you can see if the calculator has tried to plot the x -coordinate of the hole. If it has not, then the hole will remain invisible as the calculator connects one point to the next and will thus “go over top” of the hole. Using Zoom Decimal on the TI-83 may help to “see” the hole. Also using the table of values can help check for holes in the graph.
For example, the graph of 
, drawn using ZOOM STANDARD, in DOT MODE does not reveal the hole at the point 
.
The graph of the same function using ZOOM DECIMAL in DOT MODE does reveal the hole.
Using a WINDOW of 
and 
will also reveal the hole in the example above. TRACING using the above window causes jumps of 0.2 in the x value.
In most cases, the students should be determining where they feel discontinuities should be for a given function by analyzing the expression and sketching the graphs. The graphing calculator should be used to clarify and verify graphs, but students will need to know how to interpret what they see.
A parking lot charges $2 for the first half hour of parking or portion thereof. The charge increases to $3 per hour or portion thereof for parking beyond 30 minutes up to a daily maximum of $15.00. The table below helps to clarify this statement.
Draw the graph of the function for a 24-hour time period. Locate the points of discontinuity and classify them. What are the implications for someone parking in this lot?
Have the students investigate the graph of 
to experience an oscillating discontinuity.
See Appendix A for additional objectives and activities.