Simple limit 2

Add 1 and substract 1 in numerator. Seperate trignometric ratio and exponential term. Expand exponential term and use half angle formula for 1-cos(x) .

Use L'Hopital's rule twice!

$e^{x^2} - cos x$ ~ (1 + $\frac {x^2}{1!} + \frac {x^4}{2!}$ + ...) - (1 - $\frac {x^2}{2!} + \frac{x^4}{4!}$ - ...) ~ $\frac {3 \cdot x^2}{2!}$ + o( $x^2$ ) when x $\rightarrow$ 0 $\Rightarrow$ the above limit is 3/2 when x $\rightarrow$ 0