• Engineering and Architecture
  • Management and Business Administration
  • Medicine and Allied Sciences
  • Law
  • Animation and Design
  • Media, Mass Communication and Journalism
  • Finance & Accounts
  • Computer Application and IT
  • Pharmacy
  • Hospitality and Tourism
  • Competition
  • School
  • Study Abroad
  • Arts, Commerce & Sciences
  • Learn
  • Online Courses and Certifications
JEE Main Cutoff 2025 - Qualifying Marks for NITs, IIITs, GFTIs
  1. Home
  2. JEE Main
  3. Study Material
  4. Limit Using Expansion - Practice Questions & MCQ

Limit Using Expansion - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • 109 Questions around this concept.

Solve by difficulty

QuestionMEDIUM

\mathrm{ \lim _{x \rightarrow \infty}\left(1+\frac{3}{x}+\frac{5}{x^2}\right)^x \text { equals }}

View Solution

\mathrm{\lim _{n \rightarrow \infty} \prod_{r=2}^n\left(\frac{r^3+1}{r^3-1}\right)} is
 

View Solution

\mathrm{\lim _{x \rightarrow 0} \frac{3 e^x-x^3-3 x-3}{\tan ^2 x}}

View Solution

Find  \mathrm{\lim _{x \rightarrow 0} \frac{(1+x)^{1 / m}-(1-x)^{1 / m}}{(1+x)^{1 / n}-(1-x)^{1 / n}}}

View Solution

The value of \mathrm{f(0)}  so that the fanction \mathrm{f(x)= \frac{\sqrt{1+x}-\sqrt[3]{1+x}}{x}} becomes continuous is equal to

View Solution

The integer \mathrm{n} for which \mathrm{\lim _{x \rightarrow 0}\left\{\frac{(\cos x-1)\left(\cos x-e^2\right)}{x^n}\right\}} is a finite, non-zero real number, is

 

View Solution

Evaluate \mathrm{\lim _{x \rightarrow 0} \frac{\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-x}-2 x}{\sin ^3 x}}

View Solution
UPES B.Tech Admissions 2025

Ranked #42 among Engineering colleges in India by NIRF | Highest CTC 50 LPA , 100% Placements

Apply
Amrita University B.Tech 2025

Recognized as Institute of Eminence by Govt. of India | NAAC ‘A++’ Grade | Upto 75% Scholarships

Apply

\mathrm{f(x)} is the integral of \mathrm{\frac{2 \sin x-\sin 2 x}{x^3}, x \neq 0, find \lim _{x \rightarrow 0} f^{\prime}(x)}.

View Solution

\mathrm{\lim _{n \rightarrow \infty} \cos \left[\pi \sqrt{\left(n^2+n\right)}\right] \text { when } n \text { is an integer, is equal to: } }

View Solution
JEE Main Exam's High Scoring Chapters and Topics
This free eBook covers JEE Main important chapters & topics to study just 40% of the syllabus and score up to 100% marks in the examination.
Download EBook

The integer \mathrm{ n} for which  \mathrm{\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}} is a finite non-zero number is:

View Solution

Concepts Covered - 2

Limit Using Expansion (Part 1)

Limit Using Expansion (Part 1)

Using the expansions is one of the methods to find the limits. The following expansion formulas which is also known as Taylor series, are very useful in evaluating various limits.

 \\\text { (i) } \quad e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots \ldots\\\\\text { (ii) } \quad a^{x}=1+x\left(\log _{e} a\right)+\frac{x^{2}}{2 !}\left(\log _{e} a\right)^{2}+\frac{x^{3}}{3 !}\left(\log _{e} a\right)^{3}+\ldots \ldots \ldots\\\\ \text { (iii) } \quad \log (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\ldots \ldots \ldots\\\\\text { (iv) } \quad \log (1-x)=-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}-\frac{x^{4}}{4}-\ldots \ldots \ldots\\\\\text{ (v) } \quad(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\ldots \ldots \ldots

Limit Using Expansion (Part 2)

Limit Using Expansion (Part 2)

\\\text { (vi) } \quad \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots \ldots .\\\\\text { (vii) } \quad \cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\ldots \ldots .\\\\\text { (viii) } \quad \tan x=x+\frac{x^{3}}{3}+\frac{2}{15} x^{5}+\ldots \ldots .\\\\\text { (ix) } \quad \sin ^{-1} x=x+\frac{1^{2} }{3 !}\cdot x^{3}+\frac{1^{2} \cdot 3^{2} }{5 !}\cdot x^{5}+\frac{1^{2} \cdot 3^{2} \cdot 5^{2} }{7 !} \cdot x^{7}\ldots \ldots \ldots\\\\\text { (x) } \quad \tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}+\ldots \ldots \ldots

Study it with Videos

Limit Using Expansion (Part 1)
Limit Using Expansion (Part 2)

Study other Related Concepts

Limit Using Expansion Current Topic

Limit
Left-Hand Limits and Right-Hand Limits
Algebra of Limits
Limit of Indeterminate Form and Algebraic limit
Limit of Algebraic function
Limit Using Expansion
Sandwich Theorem
Limits of Trigonometric Functions
Exponential and Logarithmic Limits
Limits of the form (1 power infinity)

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Books

Reference Books

Limit Using Expansion (Part 1)

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 2.15

Line : 22

Limit Using Expansion (Part 2)

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 2.15

Line : 30

Clear your Basics with NCERT

E-books & Sample Papers

JEE Main 2025 Paper 2 (B.Arch) Syllabus

815+ Downloads

JEE Main 2025 Math Syllabus

8607+ Downloads

Get Answer to all your questions

JEE Main Articles

JEE Main Important Formulas 2025 for Physics, Chemistry, Maths
Nov 21, 2024
Scholarship for BTech Students in India 2025 - Eligibility, Amount, How to Apply
Nov 21, 2024
JEE Main Registration 2025 Last Date Tomorrow, Direct Link at jeemain.nta.nic.in, Live Updates
Nov 21, 2024
JEE Main 2025 Exam Centres in Andhra Pradesh (AP)
Nov 21, 2024
Upcoming Engineering Exams 2025 - Latest Updates on JEE, VITEEE, BITSAT, MHT CET, State Exams
Nov 21, 2024
Engineering Entrance Exams Application Date 2025 & Details
Nov 21, 2024
MNNIT Allahabad Cutoff JEE Main 2024 - Check Previous Year Cutoff Here
Nov 21, 2024
NIT Calicut Cutoff JEE Main 2024 - Opening and Closing Rank
Nov 21, 2024

B.Tech/B.Arch Admissions OPEN

UPES B.Tech Admissions 2025
Apply

Ranked #42 among Engineering colleges in India by NIRF | Highest CTC 50 LPA , 100% Placements

Amrita University B.Tech 2025
Apply

Recognized as Institute of Eminence by Govt. of India | NAAC ‘A++’ Grade | Upto 75% Scholarships

Dayananda Sagar University B.Tech Admissions 2025
Apply

60+ Years of Education Legacy | UGC & AICTE Approved | Prestigious Scholarship Worth 6 Crores | H-CTC 35 LPA

Parul University B.Tech Admissions 2025
Apply

India's youngest NAAC A++ accredited University | NIRF rank band 151-200 | 2200 Recruiters | 45.98 Lakhs Highest Package

Lovely Professional University B.Tech Admissions 2025
Apply

India's Largest University | NAAC A++ Accredited | 100% Placement Record | Highest Package Offered : 3 Cr PA

Atlas SkillTech University | B.Tech Admissions
Apply

Hands on Mentoring and Code Coaching | Cutting Edge Curriculum with Real World Application

View All Application Forms
News and Notifications
11 hours ago :
November 05, 2024 - 04:02 PM IST :
October 29, 2024 - 11:15 AM IST :
BTech Admission 2025 Live: JEE Mains registrations closing tomorrow; VITEEE, MET exam dates out
November 21, 2024 - 03:28 PM IST
JEE Main 2025 registration ends tomorrow; changes introduced in exam this year
November 21, 2024 - 08:34 AM IST
JEE Main 2025 Live: NTA to close JEE registration on Friday; apply at jeemain.nta.nic.in, correction window
November 20, 2024 - 09:45 PM IST
BTech Exams 2025: JEE Main application correction dates out; MHT CET syllabus, VITEEE exam date
November 19, 2024 - 10:05 PM IST
JEE Mains 2025 session 1 registration closing soon; application correction window open till November 27
November 19, 2024 - 03:45 PM IST
JEE Main 2025 exam date will not be changed in case of clash with other exams; NTA answers FAQs
November 19, 2024 - 07:59 AM IST
JEE Mains 2025: Application correction facility from November 26 to 27; NTA answers FAQs
November 18, 2024 - 10:52 PM IST
Download Careers360 App
All this at the convenience of your phone

  • Regular Exam Updates

  • Best College Recommendations

  • College & Rank predictors

  • Detailed Books and Sample Papers

  • Question and Answers

Scan and download the app

OR

Joint Entrance Examination (Main)      
JEE Main 2025 Sample Papers
Back to top