Question:
Find $f(0)$, so that $f(x)=\frac{x}{1-\sqrt{1-x}}$ becomes continuous at $x=0$.
Solution:
If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ ....(1)
Given: $f(x)=\frac{x}{1-\sqrt{1-x}}$
$\Rightarrow f(x)=\frac{x(1+\sqrt{1-x})}{(1-\sqrt{1-x})(1+\sqrt{1-x})}$
$\Rightarrow f(x)=\frac{x(1+\sqrt{1-x})}{1-(1-x)}$
$\Rightarrow f(x)=(1+\sqrt{1-x})$
$\lim _{x \rightarrow 0}(1+\sqrt{1-x})=f(0) \quad$ [From eq. (1)]
$\Rightarrow f(0)=2$
So, for $f(0)=2$, the function $f(x)$ becomes continuous at $x=0$.