Understand the problem

True or false: \(\lim_{x \to 0} \frac{\sin x}{\log(1+\tan x}=0\)
Source of the problem
TIFR GS 2018 Part A, Problem 2
Topic
Analysis
Difficulty Level
Easy
Suggested Book
Real analysis, Bartle and Sherbert

Start with hints

Do you really need a hint? Try it first!

  • Observe that the following limit is of the form \(\frac 00\).
  • Do you remember that we always solve the limits of the form \(\frac 00\) and \(\frac{\infty}{\infty}\) by L’Hospital’s Rule.
  • Consider \(f(x)=sinx\) and \(g(x)=log(1+tanx)\)
  • Compute \(f ‘(x)=cosx\) and \(g ‘(x) = sec^2(x)/(1+tanx)\)
  • Compute the limit of \(f ‘(x)/g ‘(x)\)
  • Observe that the \(lim f ‘(x)=1\) and \(lim g ‘(x) =1 \).
  • Hence the value of the Limit is \(1\).
  • The statement is therefore TRUE.

Watch the video

Connected Program at Cheenta

College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

Similar Problems

4 questions from Sylow’s theorem: Qn 4

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 3

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 2

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 1

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

Arithmetical Dynamics: Part 0

Rational function \( R(z)= \frac {P(z)}{Q(z)} \) ; where P and Q are polynimials . There are some theory about fixed points . Theorem: Let \( \rho \) be the fixed point of the maps R and g be the Mobius map . Then \( gRg^{-1} \) has the same number of fixed points at...

Sum based on Probability – ISI MMA 2018 Question 24

This is an interesting and cute sum based on the concept of Arithematic and Geometric series .The problem is to find a solution of a probability sum.

System of the linear equation: ISI MMA 2018 Question 11

This is a cute and interesting problem based on System of the linear equation in linear algebra. Here we are finding the determinant value .

Application of eigenvalue in degree 3 polynomial: ISI MMA 2018 Question 14

This is a cute and interesting problem based on application of eigen values in 3 degree polynomial .Here we are finding the determinant value .

To find Trace of a given Matrix : ISI MMA 2018 Question 13

These is a cute and interesting sum where the trace of a given matrices needs to be found using a very simple but effective method

Order of rings: TIFR GS 2018 Part B Problem 12

This problem is a cute and simple application on the ring theory in the abstract algebra section. It appeared in TIFR GS 2018.