Knowing that $$ \left(1+ \frac{t}{\sqrt{n}}\right)^n e^{-t\sqrt{n}} = e^{-t^2/2} \exp\left( n\int_0^{t/\sqrt{n}} \left(\frac{t}{\sqrt{n}} - \sigma\right)^2 \frac{1}{(1+\sigma)^3}d\sigma\right), $$ I call $f(t) = e^{-t^2/2}$ and $f_n(t) = \left(1+ \frac{t}{\sqrt{n}}\right)^n e^{-t\sqrt{n}}$.
How to justify
a) $$\lim_{n \to \infty} \int_{-\sqrt[7]n}^0 (f_n(t) - f(t)) dt = 0$$
b) same question but over a different interval of integration $[-\sqrt n, -\sqrt[7]n]$ and $(-\infty, -\sqrt n]$.
Where comes this $\sqrt[7]n$ from?
Thank you!