Question

Check the continuity of

$f\left(x\right)=\{\begin{array}{cc}(1+3x{)}^{1/x}& x\ne 0\\ e^3& x=0\end{array}$

at $x=0$.

Answer 1

For $f\left(x\right)$ to be continuous at $x=0$, its limit must be equal to $e^3$ at $x=0^{-}or{0}^{+}$.

$\underset{x\to 0}{lim}(1+3x{)}^{1/x}=e^k$......[$1^{\infty }$ form]

where,

$k=\underset{x\to 0}{lim}\left(1/x\right)(1+3x-1)=\underset{x\to 0}{lim}3=3$

Hence,

$\underset{x\to 0}{lim}(1+3x{)}^{1/x}=e^k=e^3$

So, function is continuous at $x=0$.