# Whether lim x → a [ f ( x ) ⋅ p ( x ) ] is of indeterminate form and if not then evaluate the limits.

### 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 2E

(a)

To determine

## Whether limx→a[f(x)⋅p(x)] is of indeterminate form and if not then evaluate the limits.

Expert Solution

The limit function limxa[f(x)p(x)] is of the indeterminate form.

### Explanation of Solution

Given:

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

Definition used:

If 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.

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.

limxa[f(x)p(x)]=limxaf(x)1p(x)=limxaf(x)limxa1p(x)=01=00

Therefore, limxa[f(x)p(x)] is of indeterminate form.

(b)

To determine

### Whether limx→a[h(x)⋅p(x)] is of indeterminate form and if not then evaluate the limits.

Expert Solution

The value of limxa[h(x)p(x)] is .

### Explanation of Solution

Given:

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

Calculation:

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

Use product rule and find the limit as follows,

limxa[h(x)p(x)]=limxah(x)limxap(x)=1=

Therefore, limxa[h(x)p(x)] is not of indeterminate form and its limit is .

(c)

To determine

### Whether limx→a[p(x)⋅q(x)] is of indeterminate form and if not then evaluate the limit.

Expert Solution

The value of limxa[p(x)q(x)] is .

### Explanation of Solution

Given:

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

Calculation:

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

limxa[p(x)q(x)]=limxap(x)limxaq(x)==

Therefore, limxa[p(x)q(x)] is not of indeterminate form and its value is .

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