If g ( x ) = { x ( 3 e 1 / x + 4 ) 2 - e 1 / x , x ? 0 0 , x = 0 , then f ( x ) is

Question

If $g (x) =$ ${\begin{matrix} \frac{x (3 e^{1 / x} + 4)}{2 - e^{1 / x}}, & x \neq 0 \\ 0, & x = 0 \end{matrix}$ , then $f (x)$ is

Smart Achievers · Accepted Answer

(2)

$f^{1} (0) = \lim_{h \to 0} \frac{f (0 - h) - f (0)}{- h}$

$R f^{1} (0) = \lim_{h \to 0} \frac{f (0 + h) - f (0)}{h}$

$= \lim_{h \to 0} (\frac{h (3 e^{1 / h} + 4)}{2 - e^{1 / h}} - 0) (\frac{1}{h})$

$\lim_{h \to 0} (\frac{- h (3 e^{- 1 / h} + 4)}{2 - e^{- 1 / h}} - 0) (\frac{- 1}{h}) = 2$

$= \lim_{h \to 0} (\frac{3 + 4 e^{- 1 / h}}{2 e^{- 1 / h} - 1}) = - 3$

$Since L f^{1} (0) \neq R f^{1} (0)$

$∴$ $f (x) is differentiable at x = 0 . But f (x) is continuous at x = 0$

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