Solving limit without L'Hospital's rule

Question

I wonder how manipulation techniques can solve the following limit:

$$\lim_{x\to 0} \frac{e^{x^2}- \cos x}{\sin^2 x}$$

Answer 1

Note that by standard limits

$$\frac{e^{x^2}- \cos x}{\sin^2 x}=\frac{\frac{e^{x^2}-1}{x^2}+ \frac{1-\cos x}{x^2}}{\frac{\sin^2 x}{x^2}}\to\frac{1+\frac12}{1}=\frac32$$