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# Application of L’Hopital: TIFR GS 2018 Part A, Problem 2

This problem is a cute and simple application of the L’Hopital rule in the analysis section. It appeared in TIFR GS 2018.

# 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
Analysis
Easy
##### Suggested Book
Real analysis, Bartle and Sherbert

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.

# 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.

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