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Learning Objectives
- Recognize the basic limit laws.
- Use the limit laws to evaluate the limit of a function.
- Evaluate the limit of a function by factoring.
- Use the limit laws to evaluate the limit of a polynomial or rational function.
- Evaluate the limit of a function by factoring or by using conjugates.
- Evaluate the limit of a function by using the squeeze theorem.
In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits.
Evaluating Limits with the Limit Laws
The first two limit laws were stated in (Figure) and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions.
Basic Limit Results
For any real number and any constant ,
Evaluating a Basic Limit
Evaluate each of the following limits using (Figure) .
Solution
- The limit of as approaches is : .
- The limit of a constant is that constant: .
We now take a look at the limit laws, the individual properties of limits. The proofs that these laws hold are omitted here.
Limit Laws
Let and be defined for all over some open interval containing . Assume that and are real numbers such that and . Let be a constant. Then, each of the following statements holds:
Sum law for limits :
Difference law for limits :
Constant multiple law for limits :
Product law for limits :
Quotient law for limits : for
Power law for limits : for every positive integer .
Root law for limits : for all if is odd and for if is even.
We now practice applying these limit laws to evaluate a limit.
Evaluating a Limit Using Limit Laws
Use the limit laws to evaluate .
Solution
Let’s apply the limit laws one step at a time to be sure we understand how they work. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied.
Using Limit Laws Repeatedly
Use the limit laws to evaluate .
Solution
To find this limit, we need to apply the limit laws several times. Again, we need to keep in mind that as we rewrite the limit in terms of other limits, each new limit must exist for the limit law to be applied.
Use the limit laws to evaluate . In each step, indicate the limit law applied.
Hint
Begin by applying the product law.
Solution
Limits of Polynomial and Rational Functions
By now you have probably noticed that, in each of the previous examples, it has been the case that . This is not always true, but it does hold for all polynomials for any choice of and for all rational functions at all values of for which the rational function is defined.
Limits of Polynomial and Rational Functions
Let and be polynomial functions. Let be a real number. Then,
To see that this theorem holds, consider the polynomial . By applying the sum, constant multiple, and power laws, we end up with
It now follows from the quotient law that if and are polynomials for which , then
(Figure) applies this result.
Evaluating a Limit of a Rational Function
Evaluate the .
Solution
Since 3 is in the domain of the rational function , we can calculate the limit by substituting 3 for into the function. Thus,
Additional Limit Evaluation Techniques
As we have seen, we may easily evaluate the limits of polynomials and limits of some (but not all) rational functions by direct substitution. However, as we saw in the introductory section on limits, it is certainly possible for to exist when is undefined. The following observation allows us to evaluate many limits of this type:
If for all over some open interval containing , then .
To understand this idea better, consider the limit .
The function
and the function are identical for all values of The graphs of these two functions are shown in (Figure) .
We see that
The limit has the form , where and . (In this case, we say that has the indeterminate form 0/0.) The following Problem-Solving Strategy provides a general outline for evaluating limits of this type.
Problem-Solving Strategy: Calculating a Limit When has the Indeterminate Form
- First, we need to make sure that our function has the appropriate form and cannot be evaluated immediately using the limit laws.
- We then need to find a function that is equal to for all over some interval containing . To do this, we may need to try one or more of the following steps:
- If and are polynomials, we should factor each function and cancel out any common factors.
- If the numerator or denominator contains a difference involving a square root, we should try multiplying the numerator and denominator by the conjugate of the expression involving the square root.
- If is a complex fraction, we begin by simplifying it.
- Last, we apply the limit laws.
The next examples demonstrate the use of this Problem-Solving Strategy. (Figure) illustrates the factor-and-cancel technique; (Figure) shows multiplying by a conjugate. In (Figure) , we look at simplifying a complex fraction.
Evaluating a Limit by Factoring and Canceling
Evaluate .
Solution
Step 1. The function is undefined for . In fact, if we substitute 3 into the function we get 0/0, which is undefined. Factoring and canceling is a good strategy:
Step 2. For all . Therefore,
Step 3. Evaluate using the limit laws:
Evaluating a Limit by Multiplying by a Conjugate
Evaluate .
Solution
Step 1. has the form 0/0 at -1. Let’s begin by multiplying by , the conjugate of , on the numerator and denominator:
Step 2. We then multiply out the numerator. We don’t multiply out the denominator because we are hoping that the in the denominator cancels out in the end:
Step 3. Then we cancel:
Step 4. Last, we apply the limit laws:
Evaluating a Limit by Simplifying a Complex Fraction
Evaluate .
Solution
Step 1. has the form 0/0 at 1. We simplify the algebraic fraction by multiplying by :
Step 2. Next, we multiply through the numerators. Do not multiply the denominators because we want to be able to cancel the factor :
Step 3. Then, we simplify the numerator:
Step 4. Now we factor out -1 from the numerator:
Step 5. Then, we cancel the common factors of :
Step 6. Last, we evaluate using the limit laws:
(Figure) does not fall neatly into any of the patterns established in the previous examples. However, with a little creativity, we can still use these same techniques.
Evaluating a Limit When the Limit Laws Do Not Apply
Evaluate .
Solution
Both and fail to have a limit at zero. Since neither of the two functions has a limit at zero, we cannot apply the sum law for limits; we must use a different strategy. In this case, we find the limit by performing addition and then applying one of our previous strategies. Observe that
Thus,
Evaluate .
Hint
Use the same technique as (Figure) . Don’t forget to factor before getting a common denominator.
Solution
Let’s now revisit one-sided limits. Simple modifications in the limit laws allow us to apply them to one-sided limits. For example, to apply the limit laws to a limit of the form , we require the function to be defined over an open interval of the form ; for a limit of the form , we require the function to be defined over an open interval of the form . (Figure) illustrates this point.
Evaluating a One-Sided Limit Using the Limit Laws
Evaluate each of the following limits, if possible.
Solution
(Figure) illustrates the function and aids in our understanding of these limits.
- The function is defined over the interval . Since this function is not defined to the left of 3, we cannot apply the limit laws to compute . In fact, since is undefined to the left of 3, does not exist.
- Since is defined to the right of 3, the limit laws do apply to . By applying these limit laws we obtain .
In (Figure) we look at one-sided limits of a piecewise-defined function and use these limits to draw a conclusion about a two-sided limit of the same function.
Evaluating a Two-Sided Limit Using the Limit Laws
For evaluate each of the following limits:
Solution
(Figure) illustrates the function and aids in our understanding of these limits.
- Since for all in , replace in the limit with and apply the limit laws:
.
- Since for all in , replace in the limit with and apply the limit laws:
.
- Since and , we conclude that does not exist.
We now turn our attention to evaluating a limit of the form , where , where and . That is, has the form at .
Evaluating a Limit of the Form Using the Limit Laws
Evaluate .
Solution
Step 1. After substituting in , we see that this limit has the form . That is, as approaches 2 from the left, the numerator approaches -1 and the denominator approaches 0. Consequently, the magnitude of becomes infinite. To get a better idea of what the limit is, we need to factor the denominator:
Step 2. Since is the only part of the denominator that is zero when 2 is substituted, we then separate from the rest of the function:
Step 3. and . Therefore, the product of and has a limit of :
The Squeeze Theorem
The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions. The next theorem, called the squeeze theorem , proves very useful for establishing basic trigonometric limits. This theorem allows us to calculate limits by “squeezing” a function, with a limit at a point that is unknown, between two functions having a common known limit at . (Figure) illustrates this idea.
The Squeeze Theorem
Let , and be defined for all over an open interval containing . If
for all in an open interval containing and
where is a real number, then .
Applying the Squeeze Theorem
Apply the Squeeze Theorem to evaluate .
Solution
Because for all , we have for and for (if is negative the direction of the inequalities changes when we multiply). Since , from the Squeeze Theorem we obtain . The graphs of , and are shown in (Figure) .
Use the Squeeze Theorem to evaluate .
Hint
Use the fact that to help you find two functions such that is squeezed between them.
Solution
0
We now use the Squeeze Theorem to tackle several very important limits. Although this discussion is somewhat lengthy, these limits prove invaluable for the development of the material in both the next section and the next chapter. The first of these limits is . Consider the unit circle shown in (Figure) . In the figure, we see that is the -coordinate on the unit circle and it corresponds to the line segment shown in blue. The radian measure of angle θ is the length of the arc it subtends on the unit circle. Therefore, we see that for .