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December 13th, 2016, 09:57 AM  #1 
Member Joined: Oct 2016 From: Slovenia, Europe Posts: 52 Thanks: 5  L'Hôpital rule not working correctly? Well, probably it is me who is working wrongly
I am trying to define the limit of the function f(x)=sqrt(x^4+1) / x, when x goes to zero. I tried using the L'Hôpital rule, but I got 0. $\displaystyle \lim _{x\to 0}\left(\frac{\sqrt{x^4+1}}{x}\right)$ I have done this: $\displaystyle \lim _{x\to 0}\left(\frac{\frac{1}{2}\left(x^4+1\right)^{\frac{1}{2}}\cdot 4x^3}{1}\right)$ $\displaystyle x=0$ and got... $\displaystyle \lim _{x\to 0}\left(\frac{\frac{1}{2}\left(x^4+1\right)^{\frac{1}{2}}\cdot 4x^3}{1}\right)\:\:=0$ Well, I know it is wrong but why the L'Hôpital does not work??? Where have I made a mistake? Last edited by skipjack; December 13th, 2016 at 02:25 PM. 
December 13th, 2016, 10:12 AM  #2 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 550 
L'Hôpital is properly used ONLY when the limits of numerator and denominator individually are both zero or both unbounded. That is not the case in your problem. $\displaystyle \lim _{x \rightarrow 0} \sqrt{x^4 + 1} = 1 \ne 0.$ Last edited by skipjack; December 13th, 2016 at 02:25 PM. 
December 13th, 2016, 10:13 AM  #3 
Math Team Joined: Jul 2011 From: Texas Posts: 2,843 Thanks: 1484 
You cannot use L'Hôpital ... direct substitution does not yield the indeterminate form 0/0 fyi, the limit does not exist. Last edited by skipjack; December 13th, 2016 at 02:26 PM. 
December 13th, 2016, 10:15 AM  #4 
Member Joined: Oct 2016 From: Slovenia, Europe Posts: 52 Thanks: 5 
Thanks, it was quite stupid from me to not think about that. Last edited by skipjack; December 13th, 2016 at 02:26 PM. 
December 13th, 2016, 01:16 PM  #5 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,084 Thanks: 848 Math Focus: Wibbly wobbly timeywimey stuff.  No, just not experienced enough not to slip up every nowandagain. It happens to all of us. Dan Last edited by skipjack; December 13th, 2016 at 02:26 PM. 
December 13th, 2016, 08:00 PM  #6  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,770 Thanks: 626 Math Focus: Yet to find out.  Quote:
 

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