BuyFind

Single Variable Calculus: Concepts...

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

Single Variable Calculus: Concepts...

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

Solutions

Chapter 4.5, Problem 1E

(a)

To determine

Whether limxaf(x)g(x) is of indeterminate form, if not then evaluate the limit.

Expert Solution

Answer to Problem 1E

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 limxaf(x)p(x) is of indeterminate form, if not then evaluate the limit.

Expert Solution

Answer to Problem 1E

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 limxah(x)p(x) is of indeterminate form, and if not then evaluate the limit.

Expert Solution

Answer to Problem 1E

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 limxap(x)f(x) is of indeterminate form and if not then evaluate the limit.

Expert Solution

Answer to Problem 1E

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 limxap(x)q(x) is of indeterminate form and if not then evaluate the limits.

Expert Solution

Answer to Problem 1E

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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