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

$f\left(x\right)=-\sqrt{25-x^2}\Rightarrow f\left(1\right)=-\sqrt{24}$

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

Now, $f^{\prime }\left(x\right)=-\frac{1.(-2x)}{\sqrt[2]{225-x^2}}$

$\therefore$ $\underset{x\to 1}{lim}\frac{f^{\prime }\left(x\right)}{1}=f^{\prime }\left(1\right)=\frac{2}{\sqrt[2]{224}}=\frac{1}{\sqrt{6}}$