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 $\displaystyle \lim_{x\rightarrow 0 } \frac{x^{x+1} \sin(x)}{x\tan(x)}$.

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.)

October 26th, 2019, 10:02 PM  #3 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,340 Thanks: 983 Math Focus: Wibbly wobbly timeywimey stuff.  
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. 
October 27th, 2019, 08:21 AM  #5 
Senior Member Joined: Jun 2019 From: USA Posts: 376 Thanks: 202  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 $x\rightarrow 0^$.

October 27th, 2019, 08:36 AM  #6 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,340 Thanks: 983 Math Focus: Wibbly wobbly timeywimey stuff.  Sorry, but I don't know how you got that. $\displaystyle \lim_{x \to 0} \frac{\sin(x)}{x  \tan(x)}$ is unbounded. Dan Last edited by skipjack; November 2nd, 2019 at 11:36 PM. 
October 28th, 2019, 10:31 PM  #7  
Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math  Quote:
$\displaystyle \lim_{x\rightarrow 0} \frac{d\sin(x)}{dxd\tan(x)}=\lim_{x\rightarrow 0} \frac{1}{11/[ \cos^{2}(x)]}=\infty$. Last edited by skipjack; November 2nd, 2019 at 11:37 PM.  
November 2nd, 2019, 04:50 AM  #8 
Newbie Joined: Jun 2016 From: UK Posts: 8 Thanks: 8 Math Focus: Anything but stats :)  Careful here. Derivative of sin(x) is not 1.
Last edited by skipjack; November 2nd, 2019 at 11:38 PM. 
November 2nd, 2019, 05:10 AM  #9 
Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math  Not again! My mistake...
Last edited by skipjack; November 2nd, 2019 at 11:38 PM. 
November 2nd, 2019, 02:35 PM  #10 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,340 Thanks: 983 Math Focus: Wibbly wobbly timeywimey stuff.  Oops! I missed that one myself! Good catch. Dan Last edited by skipjack; November 2nd, 2019 at 11:39 PM. 

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