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If   f(x)=x \frac{e^{[x]}-2}{[x]+|x|}  then find the value of   \lim _{x \rightarrow 0} f(x)

Option: 1

1


Option: 2

0


Option: 3

2


Option: 4

Does not exist


Answers (1)

best_answer

Given   \qquad f(x)=x \frac{e^{[x]}-2}{[x]+|x|} \\

For      -1<x<0, \lim _{x \rightarrow 0^{-}} f(x)=x \frac{e^{-1}-2}{-1-x} \\

                                                            =0\left(\frac{e^{-1}-2}{-1-0}\right)=0 \\

\lim _{x \rightarrow 0^{-}} f(x)=0

\begin{aligned} & \text { For } 0<x<1, \lim _{x \rightarrow 0^{+}} f(x)=x \frac{e^0-2}{x} \\ & \lim _{x \rightarrow 0^{+}} f(x)=-1 \\ & \text { Since } \lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x) \end{aligned}

Hence, the limit value does not exist. 

Posted by

Ritika Kankaria

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