/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Evaluate the following limits. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

Evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\sin ^{2} 3 x}{x^{2}}$$

Short Answer

Expert verified
$$\lim_{x\rightarrow 0} \frac{\sin^{2}(3x)}{x^2}$$ Answer: The limit of the given expression as x approaches 0 is 18.

Step by step solution

01

Simplify the expression

Rewrite the expression as: $$\lim_{x\rightarrow 0} \frac{\sin^{2}(3x)}{x^2} = \lim_{x\rightarrow 0} \frac{(\sin(3x))^2}{x^2}$$
02

Apply L'H么pital's Rule

Since the limit is an indeterminate form of type \(\frac{0}{0}\) as \(x\rightarrow 0\), we can use L'H么pital's Rule. Apply it twice, as we have a squared function: $$\lim_{x\rightarrow 0} \frac{(\sin(3x))^2}{x^2} = \lim_{x\rightarrow 0} \frac{2\sin(3x) \cos(3x) \cdot 3}{2x}$$ After simplifying this expression: $$= \lim_{x\rightarrow 0} \frac{6\sin(3x)\cos(3x)}{x}$$
03

Apply L'H么pital's Rule again

As the limit is still an indeterminate form of type \(\frac{0}{0}\) as \(x\rightarrow 0\), we can apply L'H么pital's Rule once more: $$\lim_{x\rightarrow 0} \frac{6\sin(3x)\cos(3x)}{x} = \lim_{x\rightarrow 0} \frac{6(\cos^{2}(3x)-\sin^{2}(3x))\cdot 3}{1}$$
04

Evaluate the limit

Now the limit is no longer indeterminate, so we can directly substitute x=0 into the expression: $$\lim_{x\rightarrow 0} 18(\cos^2(3x) - \sin^2(3x)) = 18(\cos^2(0)-\sin^2(0)) = 18(1^2 - 0^2) = 18$$ Hence, the limit of the given expression as x approaches 0 is 18.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose you make a deposit of \(\$ P\) into a savings account that earns interest at a rate of \(100 \mathrm{r} \%\) per year. a. Show that if interest is compounded once per year, then the balance after \(t\) years is \(B(t)=P(1+r)^{t}\). b. If interest is compounded \(m\) times per year, then the balance after \(t\) years is \(B(t)=P(1+r / m)^{m t} .\) For example, \(m=12\) corresponds to monthly compounding, and the interest rate for each month is \(r / 12 .\) In the limit \(m \rightarrow \infty,\) the compounding is said to be continuous. Show that with continuous compounding, the balance after \(t\) years is \(B(t)=\overline{P e^{r t}}\).

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation \(a(t)=v^{\prime}(t)=g,\) where \(g=-9.8 \mathrm{m} / \mathrm{s}^{2}\). a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A payload is released at an elevation of \(400 \mathrm{m}\) from a hot-air balloon that is rising at a rate of \(10 \mathrm{m} / \mathrm{s}\).

Evaluate the following limits in terms of the parameters a and b, which are positive real numbers. In each case, graph the function for specific values of the parameters to check your results. $$\lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{x}$$

Verify the following indefinite integrals by differentiation. $$\int x^{2} \cos x^{3} d x=\frac{1}{3} \sin x^{3}+C$$

Locate the critical points of the following functions and use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither. $$f(x)=x^{3}+2 x^{2}+4 x-1$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.