Chapter 5.6, Problem 73E

### 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 ) = x m ,   g ( x ) = e n x ,   and   h ( x ) = ( ln   x ) n , where n > 0 ,   m > 0 ,   and   x → ∞ . lim x → ∞ ( ln   x ) n x m

To determine

To calculate: The comparative rates of increase of the function limx(lnx)nxm by using L’Hospital’s rule.

Explanation

Given:

The function,

limxâ†’âˆž(lnx)nxm

Where, m>0,n>0.

Formula used:

Differentiate numerator and denominator by Lâ€™ Hospitalâ€™s rule.

limxâ†’af(x)=f(a)

Calculation:

The function,

limxâ†’âˆž(lnx)nxm

Substitute direct limit in the indeterminate form âˆžâˆž, then apply Lâ€™ Hospitalâ€™s to obtain

limxâ†’âˆž(lnx)nxm=limxâ†’âˆžn(lnx)nâˆ’1Ã—1xmxmâˆ’1=limxâ†’âˆžn(lnx)nâˆ’1mxm

Again, substitute direct limit in the indeterminate form âˆžâˆž, then apply Lâ€™ Hospitalâ€™s to obtain

limxâ†’âˆžn(lnx)nâˆ’1mxm=limxâ†’âˆžn(nâˆ’1)(lnx)nâˆ’2Ã—1xm2

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