   Chapter 2.3, Problem 4E ### 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

# Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). lim x → − 1 ( x 4 − 3 x ) ( x 2 + 5 x + 3 )

To determine

To evaluate: The limit of the function limx1(x43x)(x2+5x+3).

Explanation

Limit Laws:

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

Limit law 6: limxa[f(x)]n=[limxaf(x)]n where n is a positive integer

Limit law 7: limxac=c

Limit law 8: limxax=a

Limit law 9: limxaxn=an where n is a positive integer.

Calculation:

Obtain the limit of the function by using the limit laws as shown below.

limx1(x43x)(x2+5x+3)=limx1(x43x)limx1(x2+5x+3) (by limit law 4)=[limx1x4limx1(3x)]limx1(x2+5x+3) (by limit law 2)

Apply limit law 1, and evaluate further as follows.

limx1(x43x)(x2+5x+3)=[limx1x4limx1(3x)](limx1(x2)+limx1(5x)+limx1(3))

Apply the appropriate laws and simplify further

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