/*! 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 11 Evaluate: \(\lim _{x \rightarrow... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 11

Evaluate: \(\lim _{x \rightarrow 0}\left(\frac{\sin [\cos x]}{1+[\cos x]}\right)\)

Short Answer

Expert verified
The limit of the expression as \(x\) approaches \(0\) is \(\frac{\sin 1 }{2}\).

Step by step solution

01

Identify the limit

The given limit expression is \(\lim _{x \rightarrow 0}\left(\frac{\sin [\cos x]}{1+[\cos x]}\right)\). Thus, \(x\) is approaching \(0\).
02

Determine the limit of \(\cos x\)

As \(x\) approaches \(0\), \(\cos x\) approaches \(1\), because \(\cos 0 = 1\). Therefore, we substitute \(\cos x\) with \(1\) in the original limit expression.
03

Calculate the limit

Replace \(\cos x\) with \(1\) in the function to get \(\frac{\sin [\cos x]}{1+[\cos x]}\) which becomes \(\frac{\sin 1 }{1+1} = \frac{\sin 1 }{2}\).
04

Final simplification

As \(\sin 1\) is just a constant, the expression simplifies to \(\frac{\sin 1 }{2}\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluating Limits
When solving calculus problems, evaluating limits is a crucial skill. Limits help us understand the behavior of functions as inputs approach specific values. They are foundational in calculus, leading to deeper concepts like derivatives and integrals.

To evaluate limits, follow these steps:
  • Identify the variable approaching a specific value.
  • Substitute directly if the function is continuous at the point.
  • If direct substitution leads to an indeterminate form, such as \(\frac{0}{0}\) or \(\infty/\infty\), apply limit properties.
  • An example is using the L'Hopital's Rule or algebraic manipulation to simplify the expression for evaluation.
Evaluating limits is like solving a puzzle, where each piece (rule or property) helps complete the picture of a function's behavior.
Trigonometric Limits
Trigonometric limits specifically deal with functions involving trigonometric components like \(\sin\), \(\cos\), and \(\tan\). Understanding these limits is vital, as they appear frequently in competitive exams like IIT JEE.

Famous trigonometric limits include:
  • \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
  • \(\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\)
These limits are foundational when approaching more complex problems. In our original exercise, knowing that \(\cos x\) approaches 1 as \(x\) tends to 0, simplifies the process considerably.

Applying trigonometric limits often involves:
  • Direct substitution when possible.
  • Using small-angle approximations for more complex limits.
  • Building familiarity with identities such as \(\cos^2 x + \sin^2 x = 1\) helps in manipulation.
Calculus Problems for IIT JEE
Preparing for calculus problems in competitive exams like IIT JEE requires mastering limits among other topics. Limits form the bedrock upon which much of calculus is built, making them indispensable in problem-solving scenarios encountered in such exams.

Key strategies include:
  • Practicing a wide range of limit problems, from simple to complex.
  • Using known limit results to quickly evaluate expressions.
  • Applying calculus concepts such as continuity and differentiability, which are often linked to limits.
The IIT JEE challenges often incorporate limits that require creative thinking beyond routine calculations. Mastery over limits paves the way for tackling derivatives, integrals, and series, which are integral sections of these exams.

By combining in-depth practice and conceptual understanding, students can approach IIT JEE calculus problems with greater confidence and skill.

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

Find the value of \(\lim _{x \rightarrow 0}\left\\{\frac{32}{x^{8}}\left(1-\cos \left(\frac{x^{2}}{2}\right)-\cos \left(\frac{x^{2}}{4}\right)+\cos \left(\frac{x^{2}}{2}\right) \cos \left(\frac{x^{2}}{4}\right)\right)\right\\}\)

Evaluate: \(\lim _{x \rightarrow 0}\left(\frac{\sin \\{x\\}}{\\{x\\}}\right)\), where \\{\\}\(=\) F.I.F

Let \(L=\lim _{x \rightarrow 0}\left[\frac{100 x}{\sin x}\right]\) and \(M=\lim _{x \rightarrow 0}\left[\frac{99 \sin x}{x}\right]\), find the value of \(L+M+2\).

Advanced Exponential Limit Evaluate: \(\lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}\)

Newton and Leibnitz Rule Evaluate: \(\lim _{x \rightarrow 0}\left(\frac{x-\int_{0}^{x} \cos \left(t^{2}\right) d t}{x^{3}-6 x}\right)\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

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.