Question

Find f (0), so that $f\left(x\right)=\frac{x}{1-\sqrt{1-x}}$ becomes continuous at x = 0.

Answer 1

If $f\left(x\right)$is continuous at x = 0, then $\underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$ ...(1)

Given: $f\left(x\right)=\frac{x}{1-\sqrt{1-x}}$

$$\begin{gathered} \Rightarrow f\left(x\right)=\frac{x\left(1+\sqrt{1-x}\right)}{\left(1-\sqrt{1-x}\right)\left(1+\sqrt{1-x}\right)} \\ \Rightarrow f\left(x\right)=\frac{x\left(1+\sqrt{1-x}\right)}{1-\left(1-x\right)} \\ \Rightarrow f\left(x\right)=\left(1+\sqrt{1-x}\right) \end{gathered}$$


$\underset{x\to 0}{\lim }\left(1+\sqrt{1-x}\right)=f\left(0\right)\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right]$
$\Rightarrow f\left(0\right)=2$

So, for $f\left(0\right)=2$, the function f(x) becomes continuous at x = 0.