# 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 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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