# Whether lim x → a f ( x ) g ( x ) is of indeterminate form, if not then evaluate the limit. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4.5, Problem 1E

(a)

To determine

## Whether limx→af(x)g(x) is of indeterminate form, if not then evaluate the limit.

Expert Solution

The limit function limxaf(x)g(x) is of the indeterminate form.

### Explanation of Solution

Given:

The limit functions are, limxaf(x)=0 and limxag(x)=0 .

Definition used:

“If the limit function is of the type limxaf(x)g(x) , where both f(x)0 and g(x)0 or both f(x) and g(x) , 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”,

limxaf(x)g(x)=limxaf(x)limxag(x)=00

Therefore, the limit function limxaf(x)g(x) is of the indeterminate form.

(b)

To determine

### Whether limx→af(x)p(x) is of indeterminate form, if not then evaluate the limit.

Expert Solution

The limit function limxaf(x)g(x) is not of the indeterminate form and the limit is 0.

### Explanation of Solution

Given:

The limit functions are, limxaf(x)=0 and limxap(x)= .

Calculation:

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

limxaf(x)p(x)=limxaf(x)limxap(x)=0=0

Therefore, limxaf(x)p(x) is not of the indeterminate form and its value is 0.

(c)

To determine

### Whether limx→ah(x)p(x) is of indeterminate form, and if not then evaluate the limit.

Expert Solution

The limit function limxah(x)p(x) is not of the indeterminate form and the limit is 0.

### Explanation of Solution

Given:

The limit functions are, limxah(x)=1 and limxap(x)= .

Calculation:

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

limxah(x)p(x)=limxah(x)limxap(x)=1=0

Therefore, limxah(x)p(x) is not of indeterminate form.and its value is 0.

(d)

To determine

### Whether limx→ap(x)f(x) is of indeterminate form and if not then evaluate the limit.

Expert Solution

The limit function limxap(x)f(x) is not of the indeterminate form and the limit is ± or the limit may not exist.

### Explanation of Solution

Given:

The limit functions are, limxap(x)= and limxaf(x)=0 .

Calculation:

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

limxap(x)f(x)=limxap(x)limxaf(x)=0

Therefore, limxap(x)f(x) is not of indeterminate form.

For finding value of limit function consider the following cases:

Case-1: If limxaf(x)=0+ then, limxap(x)f(x)=+ .

Case-2: If limxaf(x)=0 then,  limxap(x)f(x)= .

Case-3: If limxap(x)f(x)limxa+p(x)f(x) then the limit does not exist.

(e)

To determine

### Whether limx→ap(x)q(x) is of indeterminate form and if not then evaluate the limits.

Expert Solution

The limit function limxap(x)q(x) is of the indeterminate form.

### Explanation of Solution

Given:

The limit functions are, limxap(x)= and limxaq(x)= .

Calculation:

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

limxap(x)q(x)=limxap(x)limxaq(x)=

Therefore, limxap(x)q(x) is of indeterminate form.

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