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JEE Advanced 2006

Exam: JEE Advanced
Year: 2006
Subject: Mathematics
Chapter: Limits

JEE Advanced 2006 · Mathematics · Limits

For x>0, _ x arrow 0 (( x)^ 1 / x + ((1)/(x))^ x ) is

For

x>0, \lim _{x \rightarrow 0}

\left((\sin x)^{1 / x}+

\left(\frac{1}{x}\right)^{\sin x}\right)

is

Options

A0
B $-1$
C1
D2

Correct answer

C. 1

Step-by-step solution

Here,

\lim _{x \rightarrow 0}(\sin x)^{1 / x}+\lim _{x \rightarrow 0}\left(\frac{1}{x}\right)^{\sin x} \Rightarrow 0+\lim _{x \rightarrow 0} e^{\log\left(\frac{1}{x}\right)^{\sin x} } \quad\left\{

\begin{array}{l}\lim _{x \rightarrow 0}(\sin x)^{1 / x} \rightarrow 0 \\ \text{ as, (decimal) })^x \rightarrow 0\end{array}

\right\} \Rightarrow \quad e^{\lim _{x \rightarrow 0} \frac{\log(1 / x)}{\operatorname{cosec} x}}

On applying L Hospital's Rule, we get \Rightarrow \quad e^{\lim _{x \rightarrow 0} \frac{x\left(-\frac{1}{x^2}\right)}{-\operatorname{cosec} x \cot x}}=e^{\lim _{x \rightarrow 0} \frac{\sin x}{x} \tan x}=e^0=1

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