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# Application of L'Hopital: TIFR GS 2018 Part A, Problem 2 # Understand the problem

True or false: $\lim _{x \rightarrow 0} \frac{\sin x}{\log (1+\tan x)}=1$

Do you really need a hint? Try it first!

"Hint 1"
• 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.
• Hint 2
• 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 $f ‘(x)/g ‘(x)$
Hint 3
• 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

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.

# Understand the problem

True or false: $\lim _{x \rightarrow 0} \frac{\sin x}{\log (1+\tan x)}=1$

Do you really need a hint? Try it first!

"Hint 1"
• 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.
• Hint 2
• 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 $f ‘(x)/g ‘(x)$
Hint 3
• 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

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

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