/*! 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 1 The value of $\lim _{\mathrm{x} ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 1

The value of $\lim _{\mathrm{x} \rightarrow 0}\left[\frac{|\sin \mathrm{x}|}{|\mathrm{x}|}\right]$, (where [.] denotes greatest integer function) is (A) 0 (B) does not exists (C) \(-1\) (D) 1

Short Answer

Expert verified
Answer: (D) 1

Step by step solution

01

Analyzing the function inside the limit

We will first analyze the function inside the limit, which is \(\frac{|\sin x|}{|x|}\). Note that it is in terms of absolute values, which means that both the numerator and denominator are always positive or non-negative for any x 鈮� 0.
02

Determine the limit without the greatest integer function

Before applying the greatest integer function, let's find the limit as x approaches 0 of \(\frac{|\sin x|}{|x|}\). We know that \(\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1\) through L'Hopital's Rule, since we have 0/0 indeterminate form. Since \(|x|\) is the same as \(x\) when approaching 0, we also have \(\lim_{x \rightarrow 0} \frac{|\sin x|}{|x|} = 1\).
03

Apply greatest integer function and find the limit

Now, we are supposed to find the limit as x approaches 0 for the greatest integer function \([\frac{|\sin x|}{|x|}]\). Since we know that the limit of \(\frac{|\sin x|}{|x|}\) is 1 as x approaches 0, the value of the function inside the greatest integer function approaches 1. As the function approaches 1 (from either side), the greatest integer function will always be 1 when the input is exactly 1, and 0 when the input is slightly less than 1 (for instance, 0.9999). Therefore, the limit of the greatest integer function of this expression as x approaches 0 is 1. So, the correct answer is: (D) 1

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

For which of the following functions, Approx \(\mathrm{f}(\mathrm{x})\) exists : (A) \(\underset{x \rightarrow 1}{\text { Approx }} \frac{x^{2}-1}{|x-1|}\) (B) Approx \(\frac{2\\{x\\}-4}{[x]-3}\) (C) $\underset{x \rightarrow 0}{\operatorname{Approx}} \frac{1}{2-2^{\frac{1}{x}}}$ (D) None of these

The limit $\lim _{n \rightarrow \infty}\left(1+\frac{1}{5}\right)\left(1+\frac{1}{5^{2}}\right)\left(1+\frac{1}{5^{4}}\right) \ldots\left(1+\frac{1}{5^{2^{*}}}\right)$ is equal to (A) 0 (B) \(5 / 4\) (C) \(4 / 5\) (D) \(1 / 5\)

Assertion $(\mathbf{A}): \lim _{\mathrm{x} \rightarrow 0^{\circ}}\left(\mathrm{x}^{\mathrm{x}^{*}}-\mathrm{x}^{\mathrm{x}}\right)=-1$ Reason \((\mathbf{R}): \lim _{x \rightarrow 0^{\prime}} x^{x}(x-1)=-1\)

$\lim _{\mathrm{n} \rightarrow \infty}\left\\{\frac{7}{10}+\frac{29}{10^{2}}+\frac{133}{10^{3}}+\ldots . .+\frac{5^{\mathrm{n}}+2^{\mathrm{n}}}{10^{\mathrm{n}}}\right\\}$ equals (A) \(3 / 4\) (B) 2 (C) \(5 / 4\) (D) \(1 / 2\)

\(\lim _{x \rightarrow 0}\left\\{(1+x)^{\frac{2}{x}}\right\\}\) (where \(\\{x\\}\) denotes the fractional part of \(\mathrm{x}\) ) is equal to (A) \(\mathrm{e}^{2}-7\) (B) \(e^{2}-8\) (C) \(\mathrm{e}^{2}-6\) (D) None of these
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.