I want to check whether I've proved the following limit question correctly. Because i don't feel so.

$\displaystyle

\lim_{x \to \infty} 3^{\frac{1}{x}}$

attempt:

let $\displaystyle \epsilon >0$ and $\displaystyle a>0$

$\displaystyle x > a \implies \frac{1}{x}<\frac{1}{a} \implies 3^{\frac{1}{x}}-1<3^{\frac{1}{a}}-1$

so define $\displaystyle a=\frac{1}{\log_3(\epsilon+1)}$

then $\displaystyle x>a \implies 3^{\frac{1}{x}}-1<\epsilon \implies |3^{\frac{1}{x}}-1|<\epsilon$

so can anyone check my work on this