Chapter 5.6, Problem 70E

### 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

# Comparing Functions In Exercises 69-74, use L' Hospital’s Rule to determine the comparative rates of increase of the functions f ( x ) = m ,   g ( x ) = e n x ,   and   h ( x ) = ( ln   x ) n where n > 0 ,   m > 0 ,   and   x → ∞ . lim x → ∞ x 3 e 2 x

To determine

To calculate: The comparative rates of increase of the function limxx3e5x, using L’ Hospital’s Rule if necessary.

Explanation

Given:

The provided function is:

limxâ†’âˆžx3e5x.

Formula used:

Lâ€™ Hospitalâ€™s Formula:

limxâ†’cf(x)g(x)=limxâ†’cfâ€²(x)gâ€²(x)

Calculation:

Rewrite the provided function:

y=limxâ†’âˆžx3e5x

Now Apply Lâ€™ Hospitalâ€™s Rule for the comparative rates as shown below:

limxâ†’cf(x)g(x)=limxâ†’cfâ€²(x)gâ€²(x)

After simplification it would results in:

y=limxâ†’âˆžx3e5x=limxâ†’âˆž[ddx(x3)ddx(e5x)]=limxâ†’âˆž[3x25e5x]=[3(âˆž)25e5(âˆž)]

Therefore,

y=âˆž

As it again comes indeterminate form second iteration of Lâ€™ Hospitalâ€™s Rule for the comparative rates as shown below:

y=limxâ†’âˆž[</

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