Chapter 2.3, Problem 18E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Evaluate the limit, if it exists. lim h → 0 ( 2 + h ) 3 − 8 h

To determine

To evaluate: The limit of the function limh0(2+h)38h.

Explanation

Limit Laws:

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

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.

Let f(h)=(2+h)38h (1)

Note 1:

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

f(0)=(2+0)380=880=00

Note 2:

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

limh0(h)=0 (by limit law 8)

Note 3:

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

Calculation:

By note 3, take the limit h approaches zero but h0.

Simplify f(h) by using elementary algebra

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