Question

If $f\left(x\right)=-\sqrt{25-x^2}$ then find $\underset{x\to 1}{lim}\frac{f\left(x\right)-f\left(1\right)}{(x-1)}$

Answer 1

$f\left(x\right)=-\sqrt{25-x^2}$

$f\left(1\right)=-\sqrt{25-1}=-\sqrt{24}$

$\underset{x\to 1}{lim}\frac{f\left(x\right)-f\left(1\right)}{x-1}$

$\Rightarrow \underset{x\to 1}{lim}\frac{-\sqrt{25-x^2}+\sqrt{24}}{x-1}$

$\Rightarrow \underset{x\to 1}{lim}-\frac{1}{2\sqrt{25-x^2}}\times -2x$

$\Rightarrow \frac{1}{\sqrt{24}}$

On comparing with $\frac{1}{\sqrt{a}}$

We get $a=24$