Find f (0), so that


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


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 $x=0$.

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