The image shows a limit problem: $\underset{x\to 0}{\lim }\frac{e^{-x^2}-\cos x}{x^2}$. To solve this 0/0 indeterminate form, apply the Taylor series expansion around 0.

Expansions: $e^{-x^2}\approx 1-x^2+\frac{x^4}{2}$ and $\cos x\approx 1-\frac{x^2}{2}+\frac{x^4}{24}$.

Subtracting: Numerator ≈ $(1-x^2+\frac{x^4}{2})-(1-\frac{x^2}{2}+\frac{x^4}{24})=-\frac{x^2}{2}+\frac{11}{24}{x}^4$.

Divide by $x^2$: $\frac{-\frac{1}{2}+\frac{11}{24}{x}^2}{1}$. As x→0, the limit is $-\frac{1}{2}$.

Final Answer: $-\frac{1}{2}$