Convergence (to zero) for PDF of normal distribution. I need to prove that the PDF converges to zero when $n\to\infty$; that is,
$$\lim_{n\to \infty}f_n(x) =\lim_{n\to\infty} \frac{1}{\sqrt{2\pi n^{-3}}}\exp\left(-\frac{(x-\frac{1}{n})^2}{2n^{-3}}\right)\to 0$$
I have tried using L'Hopital and differentiate $3$ times as someone suggested on StackOverflow but I seem to be getting $\infty\cdot0$ that is $0$ for the exponential(as $n\to\infty$) and $\infty$ for the $n^6$ in the numerator both by hand and in maple.
Any other suggestions on how to show this would be much appreciated.

$$\log f_n(x) \propto -3 \log n -\frac{(x-\frac{1}{n})^2}{2n^{-3}} \longrightarrow -\infty$$
So $$f_n(x) \longrightarrow e^{-\infty} = 0$$
