Chapter 4.5, Problem 3E

(a)

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 3E

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

Explanation of Solution

Given:

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

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 $f\left(x\right)\to \infty$ and $g\left(x\right)\to \infty$, then the limit may or may not exist, which is called as indeterminate form of differences of limits”.

Calculation:

Sum rule for limits: Limit of the sum is sum of the limits.

Use the Sum rule for limits and find the limit as follows.

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

Therefore, $\underset{x\to a}{\lim }\left[f\left(x\right)-p\left(x\right)\right]$ is $-\infty$.

(b)

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 3E

The limit function $\underset{x\to a}{\lim }\left[p\left(x\right)-q\left(x\right)\right]$ 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:

Sum rule for limits: “Limit of the sum is sum of the limits”.

Use sum rule and find the limit as follows,

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

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

(c)

To determine

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

Answer to Problem 3E

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:

Product rule for limits: “Limit of the product is product of the limits”.

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

Therefore, $\underset{x\to a}{\lim }\left[p\left(x\right)+q\left(x\right)\right]=\infty$.