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, Problem 33RE
To determine

Tofind:the limit of the function.

Expert Solution

Answer to Problem 33RE

Thelimit of the function limx1+(xx-1-1ln(x)) is 12.

Explanation of Solution

Given:

  limx1+(xx-1-1ln(x)) .

Concept used:

If the function will be in the form of indeterminate (00) which is not valid.

In this kind of situation L Hospital’s Rule.

Which state that the limit of a quotient of the functions is equal to the limit of quotient of the derivative.

Calculation:

  limx1+(xx-1-1ln(x))

The function can be written as:

  limx1+(1+1x-1-1ln(x)) .

  limx1+(1+ln(x)-x+1(x-1)ln(x)) .

  1+limx1+(ln(x)-x+1(x-1)ln(x)) .

By putting direct x=0 the function will be in the form of indeterminate (00) which is not valid.

In this kind of situation L Hospital’s Rule.

Which state that the limit of a quotient of the functions is equal to the limit of quotient of there derivative.

  1+limx1+(ln(x)-x+1(x-1)ln(x))

  1+limx1+(ddx(ln(x)-x+1)ddx((x-1)ln(x))) .

Finding derivative of numerator and denominator.

  1+limx1+(1x11+ln(x)1x) .

By putting direct x=0 again the function will be in the form of indeterminate (00) which is not valid.

this kind of situation L Hospital’s Rule.

  1+limx1+(ddx(1x1)ddx(1+ln(x)1x)).

  1+limx1+(1x21x+1x2).

By putting it limits:

  1+(111+112).

  limx1+(xx-1-1ln(x))=12.

Hence the limit of the function limx1+(xx-1-1ln(x)) is 12.

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