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The value of f(0), so that function f(x)=\frac{\sqrt{a^{2}-ax+x^{2}}-\sqrt{a^{2}+ax+x^{2}}}{\sqrt{a+x}-\sqrt{a-x}}(a>0)  becomes continuous for all x, is given by -

Option: 1

a\sqrt{a}


Option: 2

\sqrt{a}


Option: 3

-\sqrt{a}


Option: 4

-a\sqrt{a}


Answers (1)

best_answer

 

Method of Rationalisation -

Rationalisation method is used when we have RADICAL SIGNS in an expression.(like  1/2,  1/3 etc) and there exists a negative sign between two terms of an algebraic expression.

- wherein

\lim_{x\rightarrow a}\:\frac{x-a}{\sqrt{x}-\sqrt{a}}


\therefore \:\frac{(x-a)(\sqrt{x}+\sqrt{a})}{(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})}


=\sqrt{x}+\sqrt{a}

=\sqrt{a}+\sqrt{a}

=2\sqrt{a}

 

 

\lim_{x\rightarrow 0}\frac{(a^{2}-ax+x^{2}-a^{2}-ax-x^{2})}{(a+x-a+x)}\times \frac{(\sqrt{a+x}+\sqrt{a-x})}{(\sqrt{a^{2}-ax+x^{2}}+\sqrt{a^{2}+ax+x^{2}})}

=\lim_{x\rightarrow 0}-\frac{2ax}{2x} \left ( \frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a^{2}-ax+x^{2}}+\sqrt{a^{2}+ax+x^{2}}} \right )

=-a\frac{\sqrt{a}}{a}=-\sqrt{a}

 

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Shailly goel

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