# Does a_n={(3/n)^(1/n)} converge? If so what is the limit?

Mar 27, 2016

It converges to $1$.

#### Explanation:

$f \left(x\right) = {\left(\frac{3}{x}\right)}^{\frac{1}{x}} = {e}^{\ln \frac{\frac{3}{x}}{x}}$

${\lim}_{x \rightarrow \infty} \left(\ln \frac{\frac{3}{x}}{x}\right)$ has initial form $\frac{- \infty}{\infty}$ which is indeterminate, so use l'Hospital's rule.

${\lim}_{x \rightarrow \infty} \frac{\frac{x}{3} \left(- \frac{3}{x} ^ 2\right)}{1} = {\lim}_{x \rightarrow 0} - \frac{1}{x} = 0$

Since the exponent goes to $0$, we have

${\lim}_{x \rightarrow \infty} \left(\ln \frac{\frac{3}{x}}{x}\right) = {e}^{0} = 1$