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 3E

(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 of limxa[f(x)p(x)] is

Explanation of Solution

Given:

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

Definition used:

“If limit function is of the type limxa[f(x)g(x)] , where both f(x) and g(x) , 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.

limxa[f(x)p(x)]=limxaf(x)limxap(x)=0=

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

(b)

To determine

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

Expert Solution

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

Explanation of Solution

Given:

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

Calculation:

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

Use sum rule and find the limit as follows,

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

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

(c)

To determine

Whether limx→a[p(x)+q(x)] of indeterminate form and if not then evaluate the limits.

Expert Solution

The limit 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)]= .

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