Chapter 4.5, Problem 1E

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) $\underset{x\to a}{\lim }\frac{f(x)}{g(x)}$

(b) $\underset{x\to a}{\lim }\frac{f(x)}{p(x)}$

(c) $\underset{x\to a}{\lim }\frac{h(x)}{p(x)}$

(d) $\underset{x\to a}{\lim }\frac{p(x)}{f(x)}$

(e) $\underset{x\to a}{\lim }\frac{p(x)}{q(x)}$

To determine

Whether $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is of indeterminate form, if not then evaluate the limit.

Answer to Problem 1E

The limit function $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ 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 the 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”.

Calculation:

Use the Quotient rule for limits, “limit of the quotient is quotient of the limits”,

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

Therefore, the limit function $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is of the indeterminate form.

To determine

Whether $\underset{x\to a}{\lim }\frac{f\left(x\right)}{p\left(x\right)}$ is of indeterminate form, if not then evaluate the limit.

Answer to Problem 1E

The limit function $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is not of the indeterminate form and the limit 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:

Use the Quotient rule for limits, “limit of the quotient is quotient of the limits”,

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

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

To determine

Whether $\underset{x\to a}{\lim }\frac{h\left(x\right)}{p\left(x\right)}$ is of indeterminate form, and if not then evaluate the limit.

Answer to Problem 1E

The limit function $\underset{x\to a}{\lim }\frac{h\left(x\right)}{p\left(x\right)}$ is not of the indeterminate form and the limit is 0.

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:

Use the Quotient rule for limits, “limit of the quotient is quotient of the limits”,

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

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

To determine

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

Answer to Problem 1E

The limit function $\underset{x\to a}{\lim }\frac{p\left(x\right)}{f\left(x\right)}$ is not of the indeterminate form and the limit is $\pm \infty$ or the limit may not exist.

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:

Use the Quotient rule for limits, “limit of the quotient is quotient of the limits”,

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

Therefore, $\underset{x\to a}{\lim }\frac{p\left(x\right)}{f\left(x\right)}$ is not of indeterminate form.

For finding value of limit function consider the following cases:

Case-1: If $\underset{x\to a}{\lim }f\left(x\right)=0^{+}$ then, $\underset{x\to a}{\lim }\frac{p\left(x\right)}{f\left(x\right)}={\infty }^{+}$ .

Case-2: If $\underset{x\to a}{\lim }f\left(x\right)=0^{-}$ then,  $\underset{x\to a}{\lim }\frac{p\left(x\right)}{f\left(x\right)}={\infty }^{-}$.

Case-3: If $\underset{x\to a^{-}}{\lim }\frac{p\left(x\right)}{f\left(x\right)}\ne \underset{x\to a^{+}}{\lim }\frac{p\left(x\right)}{f\left(x\right)}$ then the limit does not exist.

To determine

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

Answer to Problem 1E

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

Use the Quotient rule for limits, “limit of the quotient is quotient of the limits”,

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

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