Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 2

Suppose that
$\displaystyle \lim_{x \to a} f(x) = 0 \quad \lim_{x \to a} \quad \lim_{x \to a} h(x) = 1$
$\displaystyle \lim_{x \to a} p(x) = \infty \quad \lim_{x \to a} q(x) = \infty$

Which of the following limits are indeterminate form? Evaluate the limit if possible, for those that are not an indefinite form.

a.) $\displaystyle \lim_{x \to a} [f(x) p(x)]$
b.) $\displaystyle \lim_{x \to a} [h(x) p(x)]$
c.) $\displaystyle \lim_{x \to a} [p(x) q(x)]$


$
\begin{equation}
\begin{aligned}
\text{a.) } \lim_{x \to a} [f(x) p(x)] &= \lim_{x \to a} f(x) \cdot \lim_{x \to a} p(x)\\
\\
&= 0 \cdot \infty\\
\\
&= \frac{1}{\infty} \cdot \infty\\
\\
&= \frac{\infty}{\infty} (\text{indeterminate})\\
\\
\text{b.) } \lim_{x \to a} [h(x) p(x)] &= \lim_{x \to a} h(x) \cdot \lim_{x \to a} p(x)\\
\\
&= 1 \cdot \infty\\
\\
&= \infty\\
\\
\text{c.) } \lim_{x \to a} [p(x) q(x)] &= \lim_{x \to a} p(x) \cdot \lim_{x \to a} q(x)\\
\\
&= \infty \cdot \infty\\
\\
&= \infty
\end{aligned}
\end{equation}
$