My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Thanks Tree7Thanks
  • 1 Post By DarnItJimImAnEngineer
  • 1 Post By topsquark
  • 1 Post By DarnItJimImAnEngineer
  • 1 Post By topsquark
  • 1 Post By idontknow
  • 2 Post By RDKGames
Reply
 
LinkBack Thread Tools Display Modes
October 26th, 2019, 11:29 AM   #1
Senior Member
 
Joined: Dec 2015
From: Earth

Posts: 823
Thanks: 113

Math Focus: Elementary Math
A limit

Evaluate $\displaystyle \lim_{x\rightarrow 0 } \frac{x^{x+1} -\sin(x)}{x-\tan(x)}$.
idontknow is offline  
 
October 26th, 2019, 05:24 PM   #2
Senior Member
 
Joined: Jun 2019
From: USA

Posts: 376
Thanks: 202

Does not converge. (Goes towards $+\infty$ from the right and $-\infty$ from the left.)
Thanks from topsquark
DarnItJimImAnEngineer is offline  
October 26th, 2019, 10:02 PM   #3
Math Team
 
topsquark's Avatar
 
Joined: May 2013
From: The Astral plane

Posts: 2,340
Thanks: 983

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
Originally Posted by DarnItJimImAnEngineer View Post
Does not converge. (Goes towards $+\infty$ from the right and $-\infty$ from the left.)
I'm going to poke my nose in here. How do you handle the limit for negative x as $\displaystyle x^{x + 1}$ is a bit psychotic for negative x?

-Dan
Thanks from idontknow
topsquark is offline  
October 27th, 2019, 12:37 AM   #4
Senior Member
 
Joined: Dec 2015
From: Earth

Posts: 823
Thanks: 113

Math Focus: Elementary Math
$\displaystyle x\rightarrow 0$.
$\displaystyle l=\lim_{x\rightarrow 0 } \frac{x^{x+1} -\sin(x)}{x-\tan(x)}=\lim x^{x+1}(x-\tan(x))^{-1} - \lim \sin(x)/[x-\tan(x)]=\lim x^{x+1}/[x-\tan(x)]-3$.

$\displaystyle l=\lim -\frac{1}{\sin(x)}-3=-\infty$.

Last edited by skipjack; November 2nd, 2019 at 11:34 PM.
idontknow is offline  
October 27th, 2019, 08:21 AM   #5
Senior Member
 
Joined: Jun 2019
From: USA

Posts: 376
Thanks: 202

Quote:
Originally Posted by topsquark View Post
I'm going to poke my nose in here. How do you handle the limit for negative x?
Very lazily. I let MATLAB handle the calculations, and the real portion was antisymmetric (i.e., odd). On inspection, the imaginary part, as chosen by MATLAB, goes infinite positive as $x\rightarrow 0^-$.
Thanks from topsquark
DarnItJimImAnEngineer is offline  
October 27th, 2019, 08:36 AM   #6
Math Team
 
topsquark's Avatar
 
Joined: May 2013
From: The Astral plane

Posts: 2,340
Thanks: 983

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
Originally Posted by idontknow View Post
$\displaystyle \lim \sin(x)/[x-\tan(x)] = 3$.
Sorry, but I don't know how you got that. $\displaystyle \lim_{x \to 0} \frac{\sin(x)}{x - \tan(x)}$ is unbounded.

-Dan
Thanks from idontknow

Last edited by skipjack; November 2nd, 2019 at 11:36 PM.
topsquark is offline  
October 28th, 2019, 10:31 PM   #7
Senior Member
 
Joined: Dec 2015
From: Earth

Posts: 823
Thanks: 113

Math Focus: Elementary Math
Quote:
Originally Posted by topsquark View Post
Sorry, but I don't know how you got that. $\displaystyle \lim_{x \to 0} \frac{\sin(x)}{x - \tan(x)}$ is unbounded.

-Dan
My mistake:

$\displaystyle \lim_{x\rightarrow 0} \frac{d\sin(x)}{dx-d\tan(x)}=\lim_{x\rightarrow 0} \frac{1}{1-1/[ \cos^{2}(x)]}=\infty$.
Thanks from topsquark

Last edited by skipjack; November 2nd, 2019 at 11:37 PM.
idontknow is offline  
November 2nd, 2019, 04:50 AM   #8
Newbie
 
Joined: Jun 2016
From: UK

Posts: 8
Thanks: 8

Math Focus: Anything but stats :)
Smile

Quote:
Originally Posted by idontknow View Post
My mistake:

$\displaystyle \lim_{x\rightarrow 0} \frac{d\sin(x)}{dx-d\tan(x)}=\lim_{x\rightarrow 0} \frac{1}{1-1/[ \cos^{2}(x)]}=\infty$.
Careful here. Derivative of sin(x) is not 1.
Thanks from topsquark and idontknow

Last edited by skipjack; November 2nd, 2019 at 11:38 PM.
RDKGames is offline  
November 2nd, 2019, 05:10 AM   #9
Senior Member
 
Joined: Dec 2015
From: Earth

Posts: 823
Thanks: 113

Math Focus: Elementary Math
Quote:
Originally Posted by RDKGames View Post
Careful here. Derivative of sin(x) is not 1.
Not again! My mistake...

Last edited by skipjack; November 2nd, 2019 at 11:38 PM.
idontknow is offline  
November 2nd, 2019, 02:35 PM   #10
Math Team
 
topsquark's Avatar
 
Joined: May 2013
From: The Astral plane

Posts: 2,340
Thanks: 983

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
Originally Posted by RDKGames View Post
Careful here. Derivative of sin(x) is not 1.
Oops! I missed that one myself!

Good catch.

-Dan

Last edited by skipjack; November 2nd, 2019 at 11:39 PM.
topsquark is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
limit



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
upper limit = lower limit implies convergence zylo Calculus 13 May 31st, 2017 01:53 PM
Limit Superior and Limit Inferior veronicak5678 Real Analysis 4 August 22nd, 2011 11:07 AM
new limit panky Calculus 1 August 8th, 2011 04:14 PM
limit panky Calculus 9 July 22nd, 2011 05:11 PM
when should we evaluate left limit and right limit? conjecture Calculus 1 July 24th, 2008 02:14 PM





Copyright © 2019 My Math Forum. All rights reserved.