My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

LinkBack Thread Tools Display Modes
September 21st, 2015, 08:19 AM   #1
Senior Member
hyperbola's Avatar
Joined: Dec 2014
From: The Asymptote

Posts: 142
Thanks: 6

Math Focus: Certainty
Limit proofs

I have to limits which I wish to prove

1. $\displaystyle \lim_{x\to \infty} \frac{e^x}{x^n} = \infty$. Prove that for any positive integer n. The exponential function approaches infinity faster than any power of x.

2. $\displaystyle \lim_{x\to \infty} \frac{lnx}{x^p} = 0$. Prove that for any number p > 0. The logarithmic function approaches infinity more slowly than any power of x.

How to do so mathematically without the need to plug in values???
hyperbola is offline  
September 21st, 2015, 01:49 PM   #2
Global Moderator
Joined: May 2007

Posts: 6,451
Thanks: 567

For 1, L'Hopital's rule ends up with $\displaystyle lim_{x\to \infty }\frac{e^x}{n!}$.
For 2, L'Hopital's rule gives $\displaystyle lim_{x\to \infty }\frac {\frac{1}{x}}{px^{p-1}}=\frac{1}{px^p}$.
mathman is offline  
September 21st, 2015, 02:14 PM   #3
Math Team
Joined: Dec 2013
From: Colombia

Posts: 7,229
Thanks: 2411

Math Focus: Mainly analysis and algebra
If you write $\displaystyle 0 \lt \frac1t \lt t^{c-1}$ for $\displaystyle t \gt 1$ and $\displaystyle c \gt 0$ you can integrate with respect to $\displaystyle t$ between $\displaystyle 1$ and $\displaystyle x \gt 1$. Then raise each part of the inequality to the power $\displaystyle a \gt 0$ and divide by $\displaystyle x^b$ where $\displaystyle b \gt 0$. Finally, choose $\displaystyle c = {b \over 2a} \gt 0$ and use the squeeze theorem to prove a more general result than the second limit:
$\displaystyle \lim_{x \to \infty} {\log^a x \over x^b} = 0$
Setting $\displaystyle x = \mathrm e^y$ then gives a more general result than the first limit.
v8archie is offline  

  My Math Forum > College Math Forum > Calculus

limit, proofs

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Epsilon/Delta Limit Proofs HeartyBowl Calculus 1 February 23rd, 2012 02:14 AM
Proofs chris99191 Algebra 4 March 21st, 2011 08:26 PM
Need help with proofs twoly Algebra 1 June 5th, 2010 07:49 AM
Big-Oh proofs Selena Applied Math 6 March 21st, 2010 03:46 PM
proofs N010289 Algebra 1 October 22nd, 2007 10:24 PM

Copyright © 2018 My Math Forum. All rights reserved.