June 1st, 2017, 02:52 PM  #1 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  limit involving trig
Hi Guys, I am trying to find the best way to solve this limit. I know that it is undefined. $$\lim_{x\to\infty}\frac{\tan 8x}{x+\sin 3x}$$ I tried L'HÃ´pital, but am getting stuck. Thanks. Last edited by skipjack; June 1st, 2017 at 10:54 PM. 
June 1st, 2017, 03:23 PM  #2 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8 
With L'HÃ´pital $$\lim_{x\to\infty}\frac{8\sec^{2}8x}{1+3\cos3x}$$ Is it ok to stop here and state that the numerator oscillates between infinity and negative infinity and that the denominator is trapped between 2 and 0 and conclude that the limit is undefined? I need to put that in math terms. Last edited by skipjack; June 1st, 2017 at 10:55 PM. 
June 1st, 2017, 03:26 PM  #3 
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  
June 1st, 2017, 03:35 PM  #4  
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  Quote: Last edited by skipjack; June 1st, 2017 at 10:55 PM.  
June 1st, 2017, 03:42 PM  #5 
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  
June 1st, 2017, 03:44 PM  #6 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  Yeah, but I wanted to see the math. I am almost certain that L'HÃ´pital is involved.
Last edited by skipjack; June 1st, 2017 at 10:56 PM. 
June 1st, 2017, 04:49 PM  #7  
Math Team Joined: Jul 2011 From: Texas Posts: 2,701 Thanks: 1359  Quote:
exist and be zero or both be infinite. Have a look at the link which discusses a similar limit ... https://math.stackexchange.com/quest...doesnotexist Last edited by skipjack; June 1st, 2017 at 10:57 PM.  
June 1st, 2017, 04:55 PM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,146 Thanks: 2386 Math Focus: Mainly analysis and algebra  When $8x = 2n\pi + \frac\pi2$, the numerator is undefined, switching from arbitrarily large and positive to arbitrarily large and negative. At this moment, the denominator is positive and finite for large $x$ and so the value of the function switches from arbitrarily large and positive to arbitrarily large and negative. Thus there can be no limit.
Last edited by skipjack; June 1st, 2017 at 10:58 PM. 
June 1st, 2017, 05:02 PM  #9 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8 
Thanks a bunch, guys. It's finally clear.
Last edited by skipjack; June 1st, 2017 at 10:59 PM. 
June 1st, 2017, 07:40 PM  #10 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,146 Thanks: 2386 Math Focus: Mainly analysis and algebra 
Thinking about it, the switching from positive to negative isn't necessary. Simply the fact that for any given value of $x$ close to $\frac18(2n+\frac12)\pi$ the fraction becomes arbitrarily large is enough to deny convergence.
Last edited by skipjack; June 1st, 2017 at 10:59 PM. 

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involving, limit, trig 
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