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

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

90 STEP 4. Review the Knowledge You Need to Score High

Squeeze Theorem

If f, g, andh are functions defined on some open interval containinga such that

g(x)≤ f(x)≤h(x) for allxin the interval except possibly ataitself, and limx→ag(x)=

xlim→ah(x)=L, then limx→a f(x)=L.

Theorems on Limits

(1) limx→ 0

sinx

x

=1 and (2) limx→ 0

cosx− 1

x

= 0

Example 1

Find the limit if it exists: limx→ 0

sin 3x

x

.

Substituting 0 into the expression would lead to 0/0. Rewrite

sin 3x

x

as

3

3

·

sin 3x

x

and

thus, limx→ 0

sin 3x

x

=limx→ 0

3 sin 3x

3 x

=3 limx→ 0

sin 3x

3 x

.Asxapproaches 0, so does 3x. Therefore,

3 limx→ 0

sin 3x

3 x

=3 lim 3 x→ 0

sin 3x

3 x

=3(1)=3. (Note that lim 3 x→ 0

sin 3x

3 x

is equivalent to limx→ 0

sinx

x

by

replacing 3xbyx.) Verify your result with a calculator. (See Figure 6.1-7.)

[−10, 10] by [−4, 4]

Figure 6.1-7

Example 2

Find the limit if it exists: limh→ 0

sin 3h

sin 2h

.

Rewrite

sin 3h

sin 2h

as

3

(

sin 3h

3 h

)

2

(

sin 2h

2 h

).Ash approaches 0, so do 3h and 2h. Therefore,

hlim→ 0

sin 3h

sin 2h

=

3 lim 3 h→ 0

sin 3h

3 h

2 lim 2 h→ 0

sin 2h

2 h

=

3(1)

2(1)

=

3

2

. (Note that substitutingh=0 into the original

expression would have produced 0/0.) Verify your result with a calculator. (See Figure

6.1-8.)