   Chapter 2.3, Problem 16E

Chapter
Section
Textbook Problem

Evaluate the limit, if it exists. lim x → − 1 2 x 2 + 3 x + 1 x 2 − 2 x − 3

To determine

To evaluate: The limit of the function limx12x2+3x+1x22x3.

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 3: limxa[cf(x)]=climxaf(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.

Calculation:

The limit of the denominator is zero.

limx1(x22x3)=limx1(x2)limx1(2x)limx1(3) (by limit law 2)=limx1(x2)2limx1(x)limx1(3) (by limit law 3)=(1)22(1)3 (by limit law 9,8 and 7)=0

The quotient law cannot be used as the denominator of the function tends to zero when x1.

Let f(x)=2x2+3x+1x22x3 (1)

Simplify f(x) by using elementary algebra

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