Mathematics

This is a limit of an indeterminate form $\frac{0}{0}$, so you'll need Taylor series. The key insight is that both the numerator and denominator involve $x^2$ terms, so expand $e^{x^2}$ and $\cos x$ around $x=0$ and keep terms up to $x^2$ (or slightly higher to see the pattern). Once you subtract and simplify, the $x^2$ in the denominator will cancel with the leading term in the numerator, revealing the limit.

This is a limit of an indeterminate form $\frac{0}{0}$, so you'll need Taylor series. The key insight is that both the numerator and denominator involve $x^2$ terms, so expand $e^{x^2}$ and $\cos x$ around $x=0$ and keep terms up to $x^2$ (or slightly higher to see the pattern). Once you subtract and simplify, the $x^2$ in the denominator will cancel with the leading term in the numerator, revealing the limit.