Chapter 4.5, Problem 2E

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

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

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

To determine

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

Answer to Problem 2E

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

Explanation of Solution

Given:

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

Definition used:

If limit function is of the type $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$, where both $f\left(x\right)\to 0$ and $g\left(x\right)\to 0$ or both $f\left(x\right)\to \infty$ and $g\left(x\right)\to \infty$, then the limit may or may not exist, which is called as indeterminate form of limit.

Quotient rule for limits: “Limit of the quotient is quotient of the limits”,

Calculation:

Rearrange the given function in quotient form and then use the Quotient rule.

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

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

To determine

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

Answer to Problem 2E

The value of $\underset{x\to a}{\lim }\left[h\left(x\right)\cdot p\left(x\right)\right]$ is $\infty$.

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:

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

Use product rule and find the limit as follows,

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

Therefore, $\underset{x\to a}{\lim }\left[h\left(x\right)\cdot p\left(x\right)\right]$ is not of indeterminate form and its limit is $\infty$.

To determine

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

Answer to Problem 2E

The value of $\underset{x\to a}{\lim }\left[p\left(x\right)\cdot q\left(x\right)\right]$ 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”.

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

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