The function f:R \ 0\ arrow R given by f(x)=(1)/(x)- 2 e^ 2 x -1 can be made continuous at x=0 by defining f(0) as

Correct answer: D. 1 _ x arrow 0 (1)/(x)- 2 e^ 2 x -1 _ x arrow 0 e^ 2 x -1-2 x x (e^ 2 x -1) _ x arrow 0 2 e^ 2 x -2 (e^ 2 x -1)+2 x e^ 2 x _ x arrow 0 4 e^ 2 x 4 e^ 2 x +4 x e^ 2 x =1 .

JEE Main 2007

Exam: JEE Main
Year: 2007
Subject: Mathematics
Chapter: Limits

JEE Main 2007 · Mathematics · Limits

The function f:R \ 0\ arrow R given by f(x)=(1)/(x)- 2 e^ 2 x -1...

The function

f:R \sim\{0\} \rightarrow R

given by

f(x)=\frac{1}{x}-\frac{2}{e^{2 x}-1}

can be made continuous at

x=0

by defining

f(0)

Options

A $2$
B $-1$
C $0$
D $1$

Correct answer

D. $1$

Step-by-step solution

\lim _{x \rightarrow 0} \frac{1}{x}-\frac{2}{e^{2 x}-1} \lim _{x \rightarrow 0} \frac{e^{2 x}-1-2 x}{x

\left(e^{2 x}-1\right)}

\lim _{x \rightarrow 0} \frac{2 e^{2 x}-2}{\left(e^{2 x}-1\right)+2 x e^{2 x}}

\lim _{x \rightarrow 0} \frac{4 e^{2 x}}{4 e^{2 x}+4 x e^{2 x}}=1

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