Answer
$1-\frac{3x^{2}}{2}+\frac{25x^{4}}{24}+...$
Work Step by Step
$y=e^{-x^{2}}cosx$
$cosx=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-....$
and $e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$
Thus, $e^{-x^{2}}=1-x^{2}+\frac{x^{4}}{2!}-\frac{x^{6}}{3!}+...$
Plug into the limit to get
$e^{-x^{2}}cosx=(1-x^{2}+\frac{x^{4}}{2!}-\frac{x^{6}}{3!}+...)(1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-....)$
$\approx 1-\frac{3x^{2}}{2}+\frac{25x^{4}}{24}+...$
