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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 3 x + 6 x x 3 3 x 2

To determine

To evaluate: The limit of the function limx3x+6xx33x2.

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).

Difference of squared formula: (a2b2)=(a+b)(ab)

Fact 1:

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

Evaluation:

Let f(x)=x+6xx33x2 (1)

Note 1:

The direct substitution method is not applicable for the function f(x) as the function f(3) is in an indeterminate form when x=3.

f(3)=3+63(3)33(3)2=93273(9)=332727=00

Note 2:

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

Calculation:

By note 2, consider the limit x approaches to 3 but x3.

Simplify f(x) by using elementary algebra.

f(x)=x+6xx33x2=x+6xx2(x3)

Multiply both the numerator and the denominator by the conjugate of the numerator,

f(x)=x+6xx33x2×x+6+xx+6+x

Apply the difference of squared formula,

f(x)=(x+6)2(x)2x2(x3)(x+6+x)=x+6x2x2(x3)(x+6+x)

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