/*! 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 25 Evaluate the following limits. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 25

Evaluate the following limits. $$\lim _{x \rightarrow \pi} \frac{\cos x+1}{(x-\pi)^{2}}$$

Short Answer

Expert verified
Question: Evaluate the limit of \(f(x) = \frac{\cos x + 1}{(x - \pi)^2}\) as \(x\) approaches \(\pi\). Answer: The limit of the given function as \(x\) approaches \(\pi\) is \(\frac{1}{2}\).

Step by step solution

01

Attempt direct substitution

First, we attempt direct substitution by replacing x with 蟺 in the function: $$\frac{\cos \pi + 1}{(\pi - \pi)^2} = \frac{-1 + 1}{0^2} = \frac{0}{0}$$ Since we get the indeterminate form \(\frac{0}{0}\), we need to use another method to evaluate the limit.
02

Apply L'Hopital's Rule

Since the given limit is in the form \(\frac{0}{0}\), we can apply L'H么pital's Rule. We find the derivatives of the numerator and denominator with respect to x: $$\frac{d}{dx}(\cos x + 1) = -\sin x$$ $$\frac{d}{dx}((x - \pi)^2) = 2(x - \pi)$$ Now, we have the limit: $$\lim_{x \rightarrow \pi} \frac{-\sin x}{2(x - \pi)}$$
03

Attempt direct substitution again

Replace x with 蟺 in the new limit: $$\frac{-\sin \pi}{2(\pi - \pi)} = \frac{0}{0}$$ We still get an indeterminate form, so we apply L'Hopital's Rule again.
04

Apply L'Hopital's Rule for the second time

Find the derivatives of the numerator and denominator with respect to x for the second application of the L'Hopital's Rule: $$\frac{d}{dx} (-\sin x) = -\cos x$$ $$\frac{d}{dx} (2(x - \pi)) = 2$$ Now, the limit becomes: $$\lim_{x \rightarrow \pi} \frac{-\cos x}{2}$$
05

Evaluate the final limit

Replace x with 蟺 in the final limit: $$\lim_{x \rightarrow \pi} \frac{-\cos x}{2} = \frac{-\cos \pi}{2} = \frac{-(-1)}{2} = \frac{1}{2}$$ Thus, the limit of the given function as x approaches 蟺 is \(\frac{1}{2}\).

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

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=-32 ; v(0)=20, s(0)=0$$

Consider the functions \(f(x)=\frac{1}{x^{2 n}+1},\) where \(n\) is a positive integer. a. Show that these functions are even. b. Show that the graphs of these functions intersect at the points \(\left(\pm 1, \frac{1}{2}\right),\) for all positive values of \(n\) c. Show that the inflection points of these functions occur at \(x=\pm \sqrt[2 n]{\frac{2 n-1}{2 n+1}},\) for all positive values of \(n\) d. Use a graphing utility to verify your conclusions. e. Describe how the inflection points and the shape of the graphs change as \(n\) increases.

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

Give an argument to support the claim that if a function is concave up at a point, then the tangent line at that point lies below the curve near that point.

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty}\left(x^{2} e^{1 / x}-x^{2}-x\right)$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.