Question

The value of f (0), so that the function 

\[f\left( x \right) = \frac{\sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2}}{\sqrt{a + x} - \sqrt{a - x}}\]   becomes continuous for all x, given by

Options

• a^3/2

• a^1/2 

• −a^1/2 

• −a^3/2

Answer 1

\[- a^\frac{1}{2}\]

Given: 

\[f\left( x \right) = \frac{\sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2}}{\sqrt{a + x} - \sqrt{a - x}}\]

\[\Rightarrow f\left( x \right) = \frac{\left( \sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2} \right)\left( \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2} \right)}{\left( \sqrt{a + x} - \sqrt{a - x} \right)\left( \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2} \right)}\]
\[ \Rightarrow f\left( x \right) = \frac{\left( a^2 - ax + x^2 - \left( a^2 + ax + x^2 \right) \right)}{\left( \sqrt{a + x} - \sqrt{a - x} \right)\left( \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2} \right)}\]
\[ \Rightarrow f\left( x \right) = \frac{\left( - 2ax \right)\left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( \sqrt{a + x} - \sqrt{a - x} \right)\left( \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2} \right)\left( \sqrt{a + x} + \sqrt{a - x} \right)}\]
\[ \Rightarrow f\left( x \right) = \frac{\left( - 2ax \right)\left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( a + x - a + x \right)\left( \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2} \right)}\]
\[ \Rightarrow f\left( x \right) = \frac{\left( - 2ax \right)\left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( 2x \right)\left( \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2} \right)}\]
\[ \Rightarrow f\left( x \right) = \frac{- a\left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2} \right)}\]

If  \[f\left( x \right)\]  is continuous for all x, then it will be continuous at x = 0 as well. 

So, if  \[f\left( x \right)\]  is continuous at x = 0, then

 is continuous at x = 0, then

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

\[\Rightarrow \lim_{x \to 0} \left[ \frac{- a\left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2} \right)} \right] = f\left( 0 \right)\]
\[ \Rightarrow \left[ \frac{- 2a\left( \sqrt{a} \right)}{\left( \sqrt{a^2} + \sqrt{a^2} \right)} \right] = f\left( 0 \right)\]
\[ \Rightarrow \left[ \frac{- 2a\left( \sqrt{a} \right)}{\left( a + a \right)} \right] = f\left( 0 \right)\]
\[ \Rightarrow f\left( 0 \right) = - \sqrt{a}\]