4. For each of the following limits, say whether it is an indeterminate form, and if so,
what type of indeterminate form it is. Then find the limit in each case.
\begin{align*}
a) \lim_{x \to 0} \frac{2x + \tan x}{\sin 2x} \\
c) \lim_{x \to -\infty} x^2 e^x \\
e) \lim_{x \to 0^+} (\cot x)^x \\
g) \lim_{x \to 1} \frac{x(\ln x - 1) + 1}{(x - 1)\ln x}
\end{align*}
\begin{align*}
b) \lim_{x \to 0} \frac{\sin x - x}{x^3} \\
d) \lim_{x \to \infty} (xe^{\frac{1}{x}} - x) \\
f) \lim_{x \to \pi/2} \frac{1 - \sin x + \cos x}{\sin x + \cos x - 1}
\end{align*}
