Chapter 4.5, Problem 4E

Given that

$\begin{array}{c}\underset{x\to a}{\lim }f(x)=0\underset{x\to a}{\lim }g(x)=0\underset{x\to a}{\lim }h(x)=1\\ \underset{x\to a}{\lim }p(x)=\infty \underset{x\to a}{\lim }q(x)=\infty \end{array}$

which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.

(a) $\underset{x\to a}{\lim }{[f(x)]}^{g(x)}$

(b) $\underset{x\to a}{\lim }{[f(x)]}^{p(x)}$

(c) $\underset{x\to a}{\lim }{[h(x)]}^{p(x)}$

(d) $\underset{x\to a}{\lim }{[p(x)]}^{f(x)}$

(e) $\underset{x\to a}{\lim }{[p(x)]}^{q(x)}$

(f) $\underset{x\to a}{\lim }\sqrt[q(x)]{p(x)}$

To determine

Whether $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{g\left(x\right)}\right]$ is of indeterminate form and if not then evaluate the limits.

Answer to Problem 4E

The limit function $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{g\left(x\right)}\right]$ is of the indeterminate form.

Explanation of Solution

Given:

The limit functions are, $\underset{x\to a}{\lim }f\left(x\right)=0$ and $\underset{x\to a}{\lim }g\left(x\right)=0$.

Definition used:

“If limit function is of the type $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{g\left(x\right)}\right]$, where both $f\left(x\right)\to 0$ and $g\left(x\right)\to 0$ or both $f\left(x\right)\to \infty$ and $g\left(x\right)\to 0$ or both $f\left(x\right)\to 1$ and $g\left(x\right)\to \pm \infty$ then the limit may or may not exist, which is called as indeterminate form of differences of limits”.

Calculation:

Find the limit as follows.

$\begin{array}{c}\underset{x\to a}{\lim }\left[f{\left(x\right)}^{g\left(x\right)}\right]=\underset{x\to a}{\lim }f{\left(x\right)}^{\underset{x\to a}{\lim }g\left(x\right)}\\ =0^0\end{array}$.

Therefore, $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{g\left(x\right)}\right]$ is of indeterminate form.

To determine

Whether $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{p\left(x\right)}\right]$ is of indeterminate form and if not then evaluate the limits.

Answer to Problem 4E

The limit of $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{p\left(x\right)}\right]$ is 0.

Explanation of Solution

Given:

The limit functions are, $\underset{x\to a}{\lim }f\left(x\right)=0$ and $\underset{x\to a}{\lim }p\left(x\right)=\infty$.

Calculation:

$\begin{array}{c}\underset{x\to a}{\lim }\left[f{\left(x\right)}^{p\left(x\right)}\right]=\underset{x\to a}{\lim }f{\left(x\right)}^{\underset{x\to a}{\lim }p\left(x\right)}\\ =0^{\infty }\\ =0\end{array}$

Therefore, $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{p\left(x\right)}\right]$ is not of indeterminate form and its value is 0.

To determine

Whether $\underset{x\to a}{\lim }\left[h{\left(x\right)}^{p\left(x\right)}\right]$ is of indeterminate form and if not then evaluate the limits.

Answer to Problem 4E

The limit function $\underset{x\to a}{\lim }\left[h{\left(x\right)}^{p\left(x\right)}\right]$ is of the indeterminate form.

Explanation of Solution

Given:

The limit functions are, $\underset{x\to a}{\lim }h\left(x\right)=1$ and $\underset{x\to a}{\lim }p\left(x\right)=\infty$.

Calculation:

Rewrite the limit and find the limit as follows.

$\begin{array}{c}\underset{x\to a}{\lim }\left[h{\left(x\right)}^{p\left(x\right)}\right]=\underset{x\to a}{\lim }h{\left(x\right)}^{\underset{x\to a}{\lim }p\left(x\right)}\\ =1^{\infty }\end{array}$

Therefore, $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{g\left(x\right)}\right]$ is of the indeterminate form.

To determine

Whether $\underset{x\to a}{\lim }\left[p{\left(x\right)}^{f\left(x\right)}\right]$ is of indeterminate form and if not then evaluate the limits.

Answer to Problem 4E

The limit function $\underset{x\to a}{\lim }\left[p{\left(x\right)}^{f\left(x\right)}\right]$ is of the indeterminate form.

Explanation of Solution

Given:

The limit functions are, $\underset{x\to a}{\lim }p\left(x\right)=\infty$ and $\underset{x\to a}{\lim }f\left(x\right)=0$.

Calculation:

Find the limit as follows.

$\begin{array}{c}\underset{x\to a}{\lim }\left[p{\left(x\right)}^{f\left(x\right)}\right]=\underset{x\to a}{\lim }p{\left(x\right)}^{\underset{x\to a}{\lim }f\left(x\right)}\\ ={\infty }^0\end{array}$

Therefore, $\underset{x\to a}{\lim }\left[p{\left(x\right)}^{f\left(x\right)}\right]$ is of indeterminate form.

To determine

Whether $\underset{x\to a}{\lim }\left[p{\left(x\right)}^{q\left(x\right)}\right]$ is of indeterminate form and if not then evaluate the limits.

Answer to Problem 4E

The limit of $\underset{x\to a}{\lim }\left[p{\left(x\right)}^{q\left(x\right)}\right]$ is $\infty$.

Explanation of Solution

Given:

The limit functions are, $\underset{x\to a}{\lim }p\left(x\right)=\infty$ and $\underset{x\to a}{\lim }q\left(x\right)=\infty$.

Calculation:

The limit of the function is,

$\begin{array}{c}\underset{x\to a}{\lim }\left[p{\left(x\right)}^{q\left(x\right)}\right]=\underset{x\to a}{\lim }p{\left(x\right)}^{\underset{x\to a}{\lim }q\left(x\right)}\\ ={\infty }^{\infty }\\ =\infty \end{array}$

Therefore, $\underset{x\to a}{\lim }\left[p{\left(x\right)}^{q\left(x\right)}\right]$ is not of the indeterminate form and its value is $\infty$.

To determine

Whether $\underset{x\to a}{\lim }\left[\sqrt[q\left(x\right)]{p\left(x\right)}\right]$ is of indeterminate form and if not then evaluate the limits.

Answer to Problem 4E

The limit function $\underset{x\to a}{\lim }\left[\sqrt[q\left(x\right)]{p\left(x\right)}\right]$ is of the indeterminate form.

Explanation of Solution

Given:

The limit functions are, $\underset{x\to a}{\lim }p\left(x\right)=\infty$ and $\underset{x\to a}{\lim }q\left(x\right)=\infty$.

Calculation:

The limit is calculated as follows.

$\begin{array}{c}\underset{x\to a}{\lim }\left[\sqrt[q\left(x\right)]{p\left(x\right)}\right]=\underset{x\to a}{\lim }p{\left(x\right)}^{\underset{x\to a}{\lim }\frac{1}{q\left(x\right)}}\\ ={\infty }^0\end{array}$

Therefore, $\underset{x\to a}{\lim }\left[f{\left(x\right)}^{g\left(x\right)}\right]$ is of indeterminate form.