   Chapter 4.4, Problem 90E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

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

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# If f″ is continuous, show that lim h → 0 f ( x + h ) − 2 f ( x ) + f ( x − h ) h 2 = f ″ ( x )

To determine

To prove: If a second derivative function f(x) continuous, then limh0f(x+h)2f(x)+f(xh)h2=f(x) .

Explanation

Proof:

Obtain the value of the function as h approaches 0.

As h approaches 0 , the numerator is

f(x+h)2f(x)+f(xh)=f(x)2f(x)+f(x)=2f(x)2f(x)=00=0

As h approaches 0 , the denominator is h2=0 .

Thus, limh0f(x+h)2f(x)+f(xh)h2=00 is in an indeterminate form.

Therefore, apply L’Hospital’s Rule and obtain the limit as shown below.

limx0f(x+h)2f(x)+f(xh)h2=limh0f(x+h)f(xh)2h

Obtain the value of the function as h approaches 0 .

As h approaches 0 , the numerator is,

f(x+h)f(xh)=f(x)f(x)=00=0

And denominator is, 2h=0

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 