Question

Let $f\left(x\right)=\{\begin{array}{cc}\frac{3x+4\tan x}{x},& x\ne 0\\ k,& x=0\end{array}.$
The value of $k$, for which $f\left(x\right)$ is continuous at $x=0$ is:

Answer 1

For continuity at $x=0$, we must have $\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)=k$
$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\frac{3x+4\tan x}{x}$
$=\underset{x\to 0}{lim}(3+4\frac{\tan x}{x})=7$
$\therefore k=7$