Question

Evaluate the limit:
$\underset{x\to \infty }{lim}(a^{1/x}-1)x$

Answer 1

Given:
$\underset{x\to \infty }{lim}(a^{1/x}-1)x$

$=\underset{x\to \infty }{lim}\frac{(a^{1/x}-1)}{\frac{1}{x}}$

Put $\frac{1}{x}=t$

$=\underset{t\to 0}{lim}\frac{(a^t-1)}{t}$

$=\log a$
$[\because \underset{x\to 0}{lim}\left(\frac{a^x-1}{x}\right)=\log a]$