Question

The value of $\underset{x\to 0}{lim}{5}^{\frac{1}{x}}$ is

• A. $0$
• B. $1$
• C. $5$
• D. does not exist

Answer 1

The correct option is D does not exist

$R.H.L.=\underset{x\to 0^{+}}{lim}{5}^{\frac{1}{x}}=\infty$
$(\because \underset{x\to 0^{+}}{lim}\frac{1}{x}\to \infty )$
$L.H.L.=\underset{x\to 0^{-}}{lim}{5}^{\frac{1}{x}}=0$
$(\because \underset{x\to 0^{-}}{lim}\frac{1}{x}\to -\infty )$
$\Rightarrow L.H.L.\ne R.H.L.$

So, limit does not exist.