My Math Forum L'Hôpital rule not working correctly? Well, probably it is me who is working wrongly

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 December 13th, 2016, 10:57 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 03:25 PM.
 December 13th, 2016, 11:12 AM #2 Senior Member   Joined: May 2016 From: USA Posts: 1,192 Thanks: 489 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. $\displaystyle \lim _{x \rightarrow 0} \sqrt{x^4 + 1} = 1 \ne 0.$ Thanks from topsquark and srecko Last edited by skipjack; December 13th, 2016 at 03:25 PM.
 December 13th, 2016, 11:13 AM #3 Math Team   Joined: Jul 2011 From: Texas Posts: 2,774 Thanks: 1428 You cannot use L'Hôpital ... direct substitution does not yield the indeterminate form 0/0 fyi, the limit does not exist. Thanks from topsquark and srecko Last edited by skipjack; December 13th, 2016 at 03:26 PM.
 December 13th, 2016, 11:15 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 03:26 PM.
December 13th, 2016, 02:16 PM   #5
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 Originally Posted by srecko Thanks, it was quite stupid from me to not think about that.
No, just not experienced enough not to slip up every now-and-again. It happens to all of us.

-Dan

Last edited by skipjack; December 13th, 2016 at 03:26 PM.

December 13th, 2016, 09:00 PM   #6
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 L'Hôpital rule not working correctly?
Blame the user not the tool!

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