If function f(x)= x-a x-a , is continuous at x=a, then find value… - Vidyadip

Question

If function $f(x)=\frac{\sqrt{x}-\sqrt{a}}{x-a}$, is continuous at $x=a$, then find value of $f(a)$.

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Answer

value of $\text{R.H.L.}$
$\lim _{h \rightarrow 0} f(a+0)=\lim _{h \rightarrow 0} \frac{a+h-a}{\sqrt{a+h}-\sqrt{a}}$
$=\lim _{h \rightarrow 0}\left[\frac{h}{\sqrt{a+h}-\sqrt{a}} \times \frac{\sqrt{a+h}+\sqrt{a}}{\sqrt{a+h}+\sqrt{a}}\right]$
$=\lim _{h \rightarrow 0} h \frac{(\sqrt{a+h}+\sqrt{a})}{a+h-a}$
$=\lim _{h \rightarrow 0} \frac{h(\sqrt{a+h}+\sqrt{a})}{h}=\sqrt{a}+\sqrt{a}$
$=2 \sqrt{a}$
$\because$function is continuous at $x=a$.
${rlrl}$
$\therefore f(a) =\lim _{h \rightarrow 0} f(a+h)$
$ \therefore f(a) =2 \sqrt{a}$

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