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  

\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]       ...(1)

Given:

\[f\left( x \right) = \frac{x}{1 - \sqrt{1 - x}}\]

\[\Rightarrow f\left( x \right) = \frac{x\left( 1 + \sqrt{1 - x} \right)}{\left( 1 - \sqrt{1 - x} \right)\left( 1 + \sqrt{1 - x} \right)}\]
\[ \Rightarrow f\left( x \right) = \frac{x\left( 1 + \sqrt{1 - x} \right)}{1 - \left( 1 - x \right)}\]
\[ \Rightarrow f\left( x \right) = \left( 1 + \sqrt{1 - x} \right)\]

\[\lim_{x \to 0} \left( 1 + \sqrt{1 - x} \right) = f\left( 0 \right) \left[ \text{ From eq}  . (1) \right]\]

\[\Rightarrow f\left( 0 \right) = 2\]

So, for 

\[f\left( 0 \right) = 2\], the function f(x) becomes continuous at x = 0.