   Chapter 2.3, Problem 12E ### 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 → − 3 x 2 + 3 x x 2 − x − 12

To determine

To evaluate: The limit of the function limx3x2+3xx2x12.

Explanation

Limit Laws:

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

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

Limit law 7:limxac=c

Limit law 8:limxax=a

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.

Calculation:

Compute the limit value of the denominator.

limx3(x2x12)=limx3(x2)limx3(x)limx3(12) (by limit law 2)=(3)2(3)12 (by limit law 9,8 and 7)=9+312 (by limit law 8)=0

Since the limit of the denominator is zero, the quotient law cannot be applied.

Let f(x)=x2+3xx2x12 (1)

Simplify f(x) by using elementary algebra

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