   Chapter 2.6, Problem 43E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (a) For F ( x ) = x ln x find each of the following limits.(i) lim x → 0 + f ( x ) (ii) lim x → 1 − f ( x ) (iii) lim x → 1 + f ( x ) (b) Use a table of values to estimate lim x → ∞ f ( x ) . (c) Use the information from parts (a) and (b) to make a rough sketch of the graph of f:

(a)

To determine

To find: The values of limx0+xlnx, limx1xlnx,limx1+xlnx.

Explanation

Limit Laws used: Suppose that c is a constant and the limits limxaf(x) and limxag(x) exists, then

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

Limit law 2: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Limit law 4: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 5: limxaf(x)g(x)=limxaf(x)limxag(x) if limxag(x)0

Calculation:

Consider the function f(x)=xlnx.

Take the limit of f(x) as x approaches zero from the right.

limx0+xlnx=limx0+xlimx0+lnx=01=0

Thus, the value of the limit function is 0

(b)

To determine

To estimate: The value of limxf(x)= using the table of values.

(c)

To determine

To sketch: The graph of f using the results obtained in part (a) and (b).

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