If f(x)= 25-x^2 , then what is Lim_ x arrow 1 (f(x)-f(1))/(x-1)...

$$\begin{aligned} & f(x)=\sqrt{25-x^2} \\ & f(1)=\sqrt{24} \\ & \Rightarrow \quad \lim _{x \rightarrow 1} \frac{f(x)-f(1)}{x-1} \end{aligned}$$\therefore$It is$\frac{0}{0}$(undefined condition) so using L'hospital's rule$\Rightarrow \lim _{x \rightarrow 1} \frac{\mathrmf^{\prime}(\mathrm{x})-0}{1}=\lim _{\mathrm{x} \rightarrow 1}\left(\sqrt{25-\mathrm{x}^2}\right)^{\prime}\Rightarrow \lim _{x \rightarrow 1} \frac{1}{2} \times \frac{1}{\sqrt{25-x^2}}(-2 x)=\frac{1}{2} \times \frac{1}{\sqrt{25-(1)^2}} \times(-2)=-\frac{1}{\sqrt{24}}$