a) $ \lim_{n\rightarrow\infty}n\int_{-1}^{1}e^{(\frac{nx+1}{n})^2}-e^{-x^2}dx $
$ = \lim_{n\rightarrow\infty}\int_{-1}^{1}\frac{e^{(x+\frac{1}{n})^2}-e^{-x^2}}{\frac{1}{n}}dx $
The Mean Value Theorem implies $ \exist h \in (0,\frac{1}{n}) $ s.t. $ \frac{e^{(x+\frac{1}{n})^2}-e^{-x^2}}{\frac{1}{n}} = \frac{d}{dx}(e^{-x^2})|_{h} $
$ = \lim_{n\rightarrow\infty}\int_{-1}^{1}-2he^{-h^2}dx $
$ = \lim_{n\rightarrow\infty}-4he^{-h^2} $
as $ n\rightarrow\infty $, $ h\rightarrow 0 $ so
$ = 0 $
