Question

Evaluate $\underset{x\to 0}{lim}\frac{e^{x^2}-\cos x}{x^2}$ is $\frac{m}{n}$. Find $m+n$

Answer 1

$\underset{x\to 0}{Lt}\frac{e^{x^2}-\cos x}{x^2}=\underset{x\to 0}{L}t\frac{(e^{x^2}-1)+(1-\cos x)}{x^2}$
$=\underset{x\to 0}{Lt}\frac{e^{x^2}-1}{x^2}+\underset{x\to 0}{L}t\frac{2{\sin }^2\frac{x}{2}}{x^2}$
$=\log e+2.\frac{{\left(x/2\right)}^2}{x^2}=1+\frac{1}{2}=\frac{3}{2}$
$\Rightarrow m+n=3+2=5$