   Chapter 4.4, Problem 73E ### 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

# Prove that lim x → ∞ e x x n = ∞ for any positive integer n. This shows that the exponential function approaches infinity faster than any power of x.

To determine

To prove: For any positive integer n, the value of limxexxn=

Explanation

Proof:

Let, y=limxexxn . (1)

Take natural logarithm on both sides,

lny=ln(limxexxn)=limx(ln(exxn))=limx(lnexlnxn)=limx(xnlnx)

Rewrite as follows.

lny=limx(xnlnx)=limx(x(1nlnxx))=limxxlimx(1nlnxx)=limxx(limx1limxnlnxx)

Therefore, lny=(limxx)(1limxnlnxx) . (2)

Consider limxnlnxx and obtain its limit

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