   Chapter 2.3, Problem 24E

Chapter
Section
Textbook Problem

Evaluate the limit, if it exists. lim h → 0 ( 3 + h ) − 1 − 3 − 1 h

To determine

To evaluate: The limit of the function limh0(3+h)131h.

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(h)=(3+h)131h.

Rewrite the terms of the equation,

f(h)=1(3+h)13h (1)

Note 1:

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

f(0)=1(3+0)130=13130=00

Note 2:

The Quotient rule is not applicable for the function f(h) as 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 0 but h0

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