If f(x) = l x e^ 1/x + 1 , , , when , , , ,x 0 , , , , , , , , , … - Vidyadip

MCQ

If $f(x) = \left\{ \begin{array}{l}\frac{x}{{{e^{1/x}} + 1}},\,\,{\rm{when\,\,}}\,\,x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\,{\rm{when \,\,}}x = 0\end{array} \right.$, then

A
$\mathop {\lim }\limits_{x \to 0 + } f(x) = 1$
B
$\mathop {\lim }\limits_{x \to 0 - } f(x) = 1$
✓
$f(x)$ is continuous at $x = 0$
D
None of these

✓

Answer

Correct option: C.

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

c
(c) $f(0) = 0$; $f(0 - ) = \mathop {\lim }\limits_{h \to 0} \,\frac{{ - h}}{{{e^{ - 1/h}} + 1}} = \mathop {\lim }\limits_{h \to 0} \,\frac{{ - h}}{{1 + \frac{1}{{{e^{1/h}}}}}} = 0$

$f(0 + ) = \mathop {\lim }\limits_{h \to 0} \,\frac{h}{{{e^{1/h}} + 1}} = 0.$

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