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.

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