Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 4

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


$
\begin{equation}
\begin{aligned}
\text{a. ) } \lim_{x \to a} [f(x)]^{g(x)} &= \lim_{x \to a} [f(x)]^{\lim\limits_{x \to a}g(x)}\\
\\
&= 0^0 && \Longleftarrow \text{(Indeterminate)}\\
\\
\text{b. ) } \lim_{x \to a} [f(x)]^{p(x)} &= \lim_{x \to a} [f(x)]^{\lim\limits_{x \to a} p(x)}\\
\\
&= 0^{\infty}\\
\\
&= 0\\
\\
\text{c. ) } \lim_{x \to a} [h(x)]^{p(x)} &= \lim_{x \to a} [h(x)]^{\lim\limits_{x \to a} p(x)}\\
\\
&= 1^{\infty} && \Longleftarrow \text{(Indeterminate)}\\
\\
\text{d. ) } \lim_{x \to a} [p(x)]^{f(x)} &= \lim_{x \to a} [p(x)]^{\lim\limits_{x \to a} f(x)}\\
\\
&= \infty^0 && \Longleftarrow \text{(Indeterminate)}\\
\\
\text{e. ) } \lim_{x \to a} [p(x)]^{q(x)} &= \lim_{x \to a}[p(x)]^{\lim\limits_{x \to a} q(x)}\\
\\
&= \infty^{\infty}\\
\\
&= \infty\\
\\
\text{e. ) } \lim_{x \to a} \sqrt[q(x)]{p(x)} &= \lim_{x \to a} [p(x)]^{\frac{1}{q(x)}}\\
\\
&= \lim_{x \to a} [p(x)]^{\frac{1}{\lim\limits_{x \to a}q(x)}}\\
\\
&= \infty \frac{1}{\infty}\\
\\
&= \infty^0 && \Longleftarrow \text{(Indeterminate)}
\end{aligned}
\end{equation}
$