If f(x)= ll e^ 1 x , & when x 0 1, & when x=0 ., then - Vidyadip

MCQ

If $f(x)=\left\{\begin{array}{ll}e^{\frac{1}{x}}, & \text { when } x \neq 0 \\ 1, & \text { when } x=0\end{array}\right.$, then

A
$\lim _{x \rightarrow 0^{+}} f(x)=e$
B
$\lim _{x \rightarrow 0^{+}} f(x)=0$
✓
$f (x)$ is discontinuous at $x=0$
D
$f (x)$ is continuous at $x=0$

✓

Answer

Correct option: C.

$f (x)$ is discontinuous at $x=0$

(C)
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} e^{-1 / h}=0$
$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} e^{1 / h}=\infty$
$\therefore \quad \lim _{x \rightarrow 0^{-}} f (x) \neq \lim _{x \rightarrow 0^{+}} f (x)$
$\therefore f (x)$ is discontinuous at $x=0$.

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