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  • The value of f(0), so that the function <img alt="\mathrm{f(x)=\frac{\sqrt{a^2-a x+x^2}-\sqrt{a^2+a x+x^2}}{\sqrt{a+x}-\sqrt{a-x}} }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5C

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

Option: 1

\mathrm{a^{3 / 2} }


Option: 2

\mathrm{a^{1 / 2} }


Option: 3

\mathrm{-a^{1 / 2} }


Option: 4

\mathrm{-a^{3 / 2}


Answers (1)

best_answer

As f(x) is continuous at x=0, therefore
\mathrm{ \lim _{x \rightarrow 0} f(x)=f(0) \\ }

\mathrm{ \Rightarrow f(0)=\lim _{x \rightarrow 0} \frac{\sqrt{a^2-a x+x^2}-\sqrt{a^2+a x+x^2}}{\sqrt{a+x}-\sqrt{a-x}}}
\mathrm{ \Rightarrow f(0)=\lim _{x \rightarrow 0} \frac{-2 a x(\sqrt{a+x}+\sqrt{a-x})}{\left.\sqrt{a^2-a x+x^2}+\sqrt{a^2+a x+x^2}\right) 2 x} \\ .}

\mathrm{ =\frac{-a \cdot 2 \sqrt{a}}{2 a}=-\sqrt{a} .}
 

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shivangi.shekhar

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