Q. 62 Calculate each of the limits in ... [FREE SOLUTION]

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Chapter 3: Q. 62 (page 311)

Calculate each of the limits in Exercises 49鈥�64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.
$\lim_{x 鈫� 0} {(\sin 3 x)}^{2 x}$

Short Answer

Expert verified

$\lim_{x 鈫� 0} {(\sin 3 x)}^{2 x} = 1$

Step by step solution

Step 1. Given information

$\lim_{x 鈫� 0} {(\sin 3 x)}^{2 x}$

Step 2. Taking log on both sides

$\begin{aligned} lim_{x 鈫� 0^{+}} 鈥� \ln 鈦� (\sin 鈦� 3 x)^{2 x} = lim_{x 鈫� 0^{+}} 鈥� 2 x \ln 鈦� (\sin 鈦� 3 x) \\ = lim_{x 鈫� 0^{+}} 鈥� \frac{2 x}{\frac{1}{\ln 鈦� (\sin 鈦� 3 x)}} \\ = lim_{x 鈫� 0^{+}} 鈥� \frac{2}{鈭� \frac{\cos 鈦� 3 x 鈰� 3}{(\ln 鈦� (\sin 鈦� 3 x))^{2}} 鈰� \frac{1}{\sin 鈦� 3 x}} \\ = lim_{x 鈫� 0^{+}} 鈥� 鈭� \frac{2 (\ln 鈦� (\sin 鈦� 3 x))^{2} \sin 鈦� 3 x}{3 \cos 鈦� 3 x} \end{aligned}$

$\begin{aligned} = 鈭� lim_{x 鈫� 0^{+}} 鈥� \frac{\sin 鈦� 3 x 鈰� 2 (\ln 鈦� (\sin 鈦� 3 x))^{2}}{3 \cos 鈦� 3 x} \\ = 鈭� lim_{x 鈫� 0^{+}} 鈥� \frac{2}{3} \tan 鈦� 3 x (\ln 鈦� (\sin 鈦� 3 x))^{2} \\ = 鈭� \frac{2}{3} \tan 鈦� 0 (\ln 鈦� (\sin 鈦� 0))^{2} \\ = 0 \end{aligned}$

Step 3. Calculating the limits

$\begin{aligned} lim_{x 鈫� 0^{+}} 鈥� (\sin 鈦� 3 x)^{2 x} = e^{0} \\ = 1 \end{aligned}$

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91影视

Calculate each of the limits in Exercises 49鈥�64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.
$\lim_{x 鈫� 0} {(\sin 3 x)}^{2 x}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Taking log on both sides

Step 3. Calculating the limits

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91影视

Calculate each of the limits in Exercises 49鈥�64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.limx鈫�0(sin3x)2x

Short Answer

Step by step solution

Step 1. Given information

Step 2. Taking log on both sides

Step 3. Calculating the limits

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Geometry

Logic and Functions

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Calculate each of the limits in Exercises 49鈥�64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.
$\lim_{x 鈫� 0} {(\sin 3 x)}^{2 x}$