52 STEP 4. Review the Knowledge You Need to Score High
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 5.1-1.)
[–10, 10] by [–4, 4]
Figure 5.1-1
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 of f(x) goes higher and higher, and that asxapproaches 3 from the left, the graph
off(x) goes lower and lower. Therefore, limx→ 3
1
x− 3
is undefined. (See Figure 5.1-2.)
[–2, 8] by [–4, 4]
Figure 5.1-2
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 fbe a function and letabe a real number. Then the right-hand limit: limx→a+f(x) rep-
resents the limit of fasxapproachesafrom 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.