1. $[\mathbf{3}]$ Compute:
$$
\lim _{x \rightarrow 0} \frac{x^{2}}{1-\cos (x)}
$$
Answer: 2. Since $\sin ^{2}(x)=1-\cos ^{2}(x)$, we multiply the numerator and denominator by $1+\cos (x)$ and use the fact that $x / \sin (x) \rightarrow 1$, obtaining
$$
\lim _{x \rightarrow 0} \frac{x^{2}}{1-\cos (x)}=\lim _{x \rightarrow 0} \frac{x^{2}(1+\cos (x))}{1-\cos ^{2}(x)}=\lim _{x \rightarrow 0}\left(\frac{x}{\sin (x)}\right)^{2} \cdot 2=2
$$