Question

Solve: $\underset{x\to \infty }{lim}\left[x\right(a^{1/x}-1\left)\right],a>1$.

Answer 1

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

$=\underset{x\to \infty }{lim}\frac{(a^{1/x}-1)}{1/x}=\frac{0}{0}$ form

By L'Hospital's Rule

$=\underset{x\to \infty }{lim}\frac{-1/x^2{a}^{1/x}\log a}{-1/x^2}$

$=\underset{x\to \infty }{lim}{a}^{1/x}\log a$

$=\log a$

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