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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the limit.

lim x 1 ( 1 x 1 + 1 x 2 3 x + 2 )

To determine

To find: The limit of the function limx1(1x1+1x23x+2).

Explanation

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.

Evaluation:

Let f(x)=1x1+1x23x+2 (1)

Note 1:

The direct substitution method is not applicable for the function f(x) as the function f(1) is in an indeterminate form when x=1. That is,

f(1)=111+1123(1)+2=10+10=20

Note 2:

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

Calculation:

By note 2, consider the limit x approaches 1 but x1.

Simplify f(x) by using elementary algebra.

f(x)=1x1+1x23x+2 (2)

Factorise the denominator of f(x),

x23x+2=x22xx+2=x(x2)(x2)=(x1)(x2)

Substitute (x1)(x2) for x23x+2 in equation (2),

f(x)=1(x1)+1(x1)(x2)

Multiply and divide x2(0) in the first term of f(x)

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