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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 Lâ€™Hospitalâ€™s Rule.
• Consider $f(x)=sinx$ and $g(x)=log(1+tanx)$
• Compute $f â€˜(x)=cosx$ and $g â€˜(x) = sec^2(x)/(1+tanx)$
• Compute the limit of $f â€˜(x)/g â€˜(x)$
• Observe that the $lim f â€˜(x)=1$ and $lim g â€˜(x) =1$.
• Hence the value of the Limit is $1$.
• The statement is therefore TRUE.

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