- 2.3.1. Recognize the basic limit laws.
- 2.3.2. Use the limit laws to evaluate the limit of a function.
- 2.3.3. Evaluate the limit of a function by factoring.
- 2.3.4. Use the limit laws to evaluate the limit of a polynomial or rational function.
- 2.3.5. Evaluate the limit of a function by factoring or by using conjugates.
- 2.3.6. 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 Two Important Limits 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 a and any constant c,
We now take a look at the limit laws, the individual properties of limits. The proofs that these laws hold are omitted here.
Let and be defined for all over some open interval containing a. Assume that L and M are real numbers such that and Let c 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 n.
Root law for limits: for all L if n is odd and for if n is even and .
We now practice applying these limit laws to evaluate a limit.
Evaluating a Limit Using Limit Laws
Use the limit laws to evaluate
Using Limit Laws Repeatedly
Use the limit laws to evaluate
Use the limit laws to evaluate In each step, indicate the limit law applied.
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 a and for all rational functions at all values of a for which the rational function is defined.
Limits of Polynomial and Rational Functions
Let and be polynomial functions. Let a 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
Example 2.16 applies this result.
Evaluating a Limit of a Rational Function
Additional Limit Evaluation Techniques
As we have seen, we may evaluate easily 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 a, then
To understand this idea better, consider the limit
and the function are identical for all values of The graphs of these two functions are shown in Figure 2.24.
We see that
The limit has the form where and (In this case, we say that has the indeterminate form The following Problem-Solving Strategy provides a general outline for evaluating limits of this type.
- 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 a. 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. Example 2.17 illustrates the factor-and-cancel technique; Example 2.18 shows multiplying by a conjugate. In Example 2.19, we look at simplifying a complex fraction.
Evaluating a Limit by Factoring and Canceling
Evaluating a Limit by Multiplying by a Conjugate
Evaluating a Limit by Simplifying a Complex Fraction
Example 2.20 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
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 Example 2.21 illustrates this point.
Evaluating a One-Sided Limit Using the Limit Laws
Evaluate each of the following limits, if possible.
In Example 2.22 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:
Graph and evaluate
We now turn our attention to evaluating a limit of the form where where and That is, has the form at a.
Evaluating a Limit of the Form Using the Limit Laws
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 a that is unknown, between two functions having a common known limit at a. Figure 2.27 illustrates this idea.
The Squeeze Theorem
Let and be defined for all over an open interval containing a. If
for all in an open interval containing a and
where L is a real number, then
Applying the Squeeze Theorem
Apply the squeeze theorem to evaluate
Use the squeeze theorem to evaluate
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 2.29. In the figure, we see that is the y-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
Because and by using the squeeze theorem we conclude that
To see that as well, observe that for and hence, Consequently, It follows that An application of the squeeze theorem produces the desired limit. Thus, since and
Next, using the identity for we see that
We now take a look at a limit that plays an important role in later chapters—namely, To evaluate this limit, we use the unit circle in Figure 2.30. Notice that this figure adds one additional triangle to Figure 2.30. We see that the length of the side opposite angle θ in this new triangle is Thus, we see that for
By dividing by in all parts of the inequality, we obtain
Equivalently, we have
Since we conclude that By applying a manipulation similar to that used in demonstrating that we can show that Thus,
In Example 2.25 we use this limit to establish This limit also proves useful in later chapters.
Evaluating an Important Trigonometric Limit
Deriving the Formula for the Area of a Circle
Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287−212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit.
We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these triangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps:
- Express the height h and the base b of the isosceles triangle in Figure 2.31 in terms of and r.
- Using the expressions that you obtained in step 1, express the area of the isosceles triangle in terms of θ and r.
(Substitute for in your expression.)
- If an n-sided regular polygon is inscribed in a circle of radius r, find a relationship between θ and n. Solve this for n. Keep in mind there are 2π radians in a circle. (Use radians, not degrees.)
- Find an expression for the area of the n-sided polygon in terms of r and θ.
- To find a formula for the area of the circle, find the limit of the expression in step 4 as θ goes to zero. (Hint:
The technique of estimating areas of regions by using polygons is revisited in Introduction to Integration.
Section 2.3 Exercises
In the following exercises, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s).
In the following exercises, use direct substitution to evaluate each limit.
In the following exercises, use direct substitution to show that each limit leads to the indeterminate form Then, evaluate the limit.
where a is a non-zero real-valued constant
In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of Example 2.23 to simplify the function to help determine the limit.
In the following exercises, assume that and Use these three facts and the limit laws to evaluate each limit.
[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.
In the following exercises, use the following graphs and the limit laws to evaluate each limit.
[T] True or False? If then
For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions and when possible.
[T] In physics, the magnitude of an electric field generated by a point charge at a distance r in vacuum is governed by Coulomb’s law: where E represents the magnitude of the electric field, q is the charge of the particle, r is the distance between the particle and where the strength of the field is measured, and is Coulomb’s constant:
- Use a graphing calculator to graph given that the charge of the particle is
- Evaluate What is the physical meaning of this quantity? Is it physically relevant? Why are you evaluating from the right?
[T] The density of an object is given by its mass divided by its volume:
- Use a calculator to plot the volume as a function of density assuming you are examining something of mass 8 kg (
- Evaluate and explain the physical meaning.