/*! 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 35 If \(\lim _{x \rightarrow 0} \fr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 35

If \(\lim _{x \rightarrow 0} \frac{x^{2 n} \sin ^{n} x}{x^{2 n}-\sin ^{2 n} x}\) is a non zero finite number, then n must be equal to (A) 1 (B) 2 (C) 3 (D) none of these

Short Answer

Expert verified
Answer: (D) none of these

Step by step solution

01

Apply L'Hopital's Rule

We need to differentiate the numerator and the denominator with respect to x. So, $$ \frac{d}{d x}\left(x^{2 n} \sin ^{n} x\right) = 2nx^{2n-1}\sin^n{x} + nx^{2n}\sin^{n-1}{x} \cos{x} $$ and $$ \frac{d}{d x}\left(x^{2 n}-\sin ^{2 n} x\right) = 2nx^{2n-1} - 2n\sin^{2n-1}{x} \cos{x} $$ Since we need the limit as x approaches 0, let's put x = 0 in the derivatives: $$ \lim _{x \rightarrow 0} \frac{d\left(x^{2 n} \sin ^{n} x\right)/dx}{d\left(x^{2 n}-\sin ^{2 n} x\right)/dx} = \lim _{x \rightarrow 0} \frac{ 2nx^{2n-1}\sin^n{x} + nx^{2n}\sin^{n-1}{x} \cos{x}}{2nx^{2n-1} - 2n\sin^{2n-1}{x} \cos{x}} $$
02

Simplify the Limit Expression

To simplify the limit, let's factor out the common terms: $$ \lim _{x \rightarrow 0} \frac{n(x^{2n}(1+\sin^{2n-1}{x}\cos{x})}{2n(x^{2n-1}(1-\sin^{2n-1}{x}\cos{x})} $$ Now, cancel the common terms of n and x^(2n-1) to get: $$ \lim _{x \rightarrow 0} \frac{1 + \sin^{2n-1}{x}\cos{x}}{2(1 - \sin^{2n-1}{x}\cos{x})} = \frac{1}{2} $$ Since the limit is a non-zero finite number, our analysis shows that: n must be equal to (D) none of these.

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

Let \(l_{1}=\lim _{x \rightarrow \infty} \sqrt{\frac{x-\cos ^{2} x}{x+\sin x}}\) and $l_{2}=\lim _{\mathrm{h} \rightarrow 0^{-}} \int_{-1}^{1} \frac{\mathrm{h} \mathrm{dx}}{\mathrm{h}^{2}+\mathrm{x}^{2}} .$ Then (A) both \(l_{1}\) and \(l_{2}\) are less than \(\frac{22}{7}\) (B) one of the two limits is rational and other irrational. (C) \(l_{2}>l_{1}\) (D) \(l_{2}\) is greater than 3 times of \(l_{1}\).

Which of the following limits exist? (where [.] indicates greatest integer function all throughout) (A) \(\lim _{x \rightarrow 1} \frac{\sin [x]}{[x]}\) (B) $\lim _{n \rightarrow \infty}\left(\frac{\mathrm{e}^{\mathrm{n}}}{\pi}\right)^{1 / \mathrm{n}}$ (C) \(\lim _{x \rightarrow 1}\left[\sin \left(\sin ^{-1} x\right)\right]\) (D) \(\lim _{x \rightarrow \pi / 2}\left[\sin ^{-1}(\sin x)\right]\)

Column - I (A) \(\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{e^{x^{3}}-e^{x}+x}\) equals (B) If the value of $\lim _{\mathrm{x} \rightarrow 0^{-}}\left(\frac{(3 / \mathrm{x})+1}{(3 / \mathrm{x})-1}\right)^{1 / \mathrm{x}}$ can be expressed in the form of \(\mathrm{e}^{\mathrm{iq}}\), where \(\mathrm{p}\) and \(\mathrm{q}\) are relative prime then \((\mathrm{p}+\mathrm{q})\) is equal to (C) \(\lim _{x \rightarrow 0} \frac{\tan ^{3} x-\tan x^{3}}{x^{5}}\) equals (D) $\lim _{x \rightarrow 0} \frac{x+2 \sin x}{\sqrt{x^{2}+2 \sin x+1}-\sqrt{\sin ^{2} x-x+1}}$ Column-II (P) 1 (Q) 2 (R) 4 (S) 5

Assertion \((\mathbf{A}):\) Let \(\mathrm{f}:(0, \infty) \rightarrow \mathrm{R}\) be a twice continuously differentiable function such that $\left|f^{\prime}(x)+2 x f^{\prime}(x)+\left(x^{2}+1\right) f(x)\right| \leq 1\( for all \)x$ Then \(\lim _{x \rightarrow \infty} f(x)=0\). Reason (R) : Applying L'Hospital's rule twice on the function $\frac{f(x) e^{\frac{x^{2}}{2}}}{e^{\frac{x^{3}}{2}}}\( we get \)\lim _{x \rightarrow \infty} f(x)=0$.

If $\lim _{x \rightarrow \infty}\left(\sqrt{a^{2} x^{2}+a x+1}-\sqrt{a^{2} x^{2}+1}\right)$ $=\mathrm{K} \cdot \lim _{x \rightarrow \infty}(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})\( then the value of \)\mathrm{K}$ (A) (B) (C) \(2 \mathrm{a}\) (D) None of these
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.