Question

If $f\left(x\right)=\sqrt{25-x^2}$, then what is $\underset{x\to 1}{lim}\frac{f\left(x\right)-f\left(1\right)}{x-1}$ equal to?

• A. $\frac{1}{5}$
• B. $\frac{1}{24}$
• C. $\sqrt{24}$
• D. $-\frac{1}{\sqrt{24}}$

Answer 1

The correct option is D $-\frac{1}{\sqrt{24}}$

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

We know that, $\underset{x\to a}{lim}\frac{f\left(x\right)-f\left(a\right)}{x-a}=f^{\prime }\left(a\right)$

So, we get

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

$=\frac{f^{\prime }\left(1\right)}{1}={\left|\frac{1(-2x)}{2\sqrt{25-x^2}}\right|}_{atx=1}$

$=-\frac{1}{\sqrt{24}}$