   Chapter 2.3, Problem 19E ### 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, if it exists. lim x → − 2 x + 2 x 3 + 8

To determine

To evaluate: The limit of the function limx2x+2x3+8.

Explanation

Limit Laws:

Suppose that c is a constant and the limits limxaf(x) and limxag(x) exist, then

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

Limit law 7: limxac=c

Limit law 8: limxax=a

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

Direct substitution property:

If f is a polynomial or a rational function and a is in the domain of f, then limxaf(x)=f(a).

Fact 1:

If f(x)=g(x) when xa, then limxaf(x)=limxag(x), provided the limit exist.

Let f(x)=x+2x3+8 (1)

Note 1:

The direct substitution method is not applicable for the function f(x) since letting x=2, the function f(0) is in indeterminate form.

f(2)=(2)+2(2)3+8=2+28+8=00

Note 2:

The Quotient rule is not applicable for the function f(x) because the limit of the denominator is zero.

limx2(x3+8)=limx2(x3)+limx28 (by limit law 1)=(2)3+8 (by limit law 9 and 7)=8+8=0

Note 3:

The limit may be infinite or it may be some finite value when both the numerator and the denominator approach 0

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