Find whether $f$ is continuous at $x = 0$ or not.If $f$ is continous write 1 otherwise write 0 ?
What is the value of $f (x)$ at $x = 0$

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Solution

$f (x) = \frac{l n (x)}{\frac{1}{\sqrt{x}}}$
Applying L'Hospital's rule we get
$= \frac{\frac{1}{x}}{- \frac{1}{2 x \sqrt{x}}}$
$= - 2 \sqrt{x}$
Hence as $x \to 0$ $f (x) \to 0$ .
Thus $L H L = R H L = 0$ which is equal to the value of the $f (x)$ at $x = 0$
Hence $f (x)$ is continuous at $x = 0$

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