Chapter 5.6, Problem 116E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Evaluating a Limit Let f ( x ) = x + x sin x and g ( x ) = x 2 − 4 .(a) Show that lim x → ∞ f ( x ) g ( x ) = 0 (b) Show that lim x → ∞ f ( x ) = ∞ and lim x → ∞ g ( x ) = ∞ (c) Evaluate the limit lim x → ∞ f ' ( x ) g ' ( x ) What do you notice?(d) Do your answers to parts (a) through (c) contradict L'Hopital’s Rule? Explain your reasoning.

(a)

To determine

To prove: The expression limxf(x)g(x)=0 holds for the provided functions f(x)=x+xsinx and g(x)=x24.

Explanation

Given:

The functions f(x)=x+xsinx and g(x)=x2âˆ’4.

Formula used:

Squeeze theorem states that for three function f, g and h such that f(x)<g(x)<h(x) and limxâ†’âˆžf(x)=limxâ†’âˆžh(x)=a, then limxâ†’âˆžg(x)=a.

Proof:

Consider the left-hand side of the expression to be proved.

limxâ†’âˆžf(x)g(x)=limxâ†’âˆžx+xsinxx2âˆ’4

Divide both the numerator and denominator by x2 to obtain:

limxâ†’âˆžx+xsinxx2âˆ’4=limxâ†’âˆž(1x+<

(b)

To determine

To prove: The expressions limxf(x)= and limxg(x)= holds for the provided functions.

(c)

To determine

To calculate: The expression limxf'(x)g'(x) and to interpret it.

(d)

To determine

Whether the results obtained previously violate the L’Hospital rule or not.

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