Question

If the $f\left(x\right)=\frac{{(5^x-1)}^4}{xtan\left(x/5\right)log(1+x^2/5)}$ , $x\ne 0$ is continuous at x=0, then find $f\left(0\right)$

Answer 1

We have,

$\begin{array}{c}f\left(x\right)=\frac{{\left(5^Z-1\right)}^4}{x\tan \left(x/5\right)\log \left(1+x^2/5\right)}\\ f\left(0\right)=\frac{{\left(5^x-1\right)}^4}{x\frac{\tan \left(x/5\right)}{\frac{x}{5}}\times \frac{x}{5}\frac{\log \left(1+\frac{x^2}{5}\right)}{\frac{x^2}{5}}\times \frac{x^2}{5}}\\ f\left(0\right)={lim}_{x\to 0}\frac{{\left(5^x-1\right)}^x}{\frac{x^2}{5}\times \frac{x^2}{5}}\\ ={lim}_{x\to 0}\frac{25{\left(5^x-1\right)}^x}{x^4}\\ f\left(0\right)=25{\left(x-5\right)}^4\end{array}$