5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 7, 2018 9:52

Limits and Continuity 87

Example 6

Find the limit: limx→ 0

3 sin 2x

2 x

.

Entery 1 =

3 sin 2x

2 x

in the calculator. You see that the graph of f(x) approaches 3 asx

approaches 0. Thus, the limx→ 0

3 sin 2x

2 x

=3. (Note that had you substitutedx=0 directly

in the original expression, you would have obtained a zero in both the numerator and

denominator.) (See Figure 6.1-1.)

[−10, 10] by [−4, 4]

Figure 6.1-1

Example 7

Find the limit: limx→ 3

1

x− 3

.

Entery 1 =

1

x− 3

into your calculator. You notice that asxapproaches 3 from the right, the

graph off(x) goes higher and higher and that asxapproaches 3 from the left, the graph of

f(x) goes lower and lower. Therefore, limx→ 3

1

x− 3

is undefined. (See Figure 6.1-2.)

[−2, 8] by [−4, 4]

Figure 6.1-2

TIP • Always indicate what the final answer is, e.g., “The maximum value of f is 5.” Use
complete sentences whenever possible.

One-Sided Limits

Let f be a function and letabe a real number. Then the right-hand limit: limx→a+f(x)

represents the limit offasxapproachesafrom the right, and the left-hand limit: limx→a−f(x)

represents the limit offasxapproachesafrom the left.

Existence of a Limit

Letfbe a function and letaandLbe real numbers. Then the two-sided limit: limx→a f(x)=L

if and only if the one-sided limits exist and limx→a+f(x)=xlim→a−f(x)=L.