I already evaluated the limit, and got the answer, but I want to see if you guys can come up with an easier way to solve it : .
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Originally Posted by Chris L T521 I already evaluated the limit, and got the answer, but I want to see if you guys can come up with an easier way to solve it : . *Yawn* I'll bite. is an indeterminant form so I'll just be lazy and apply l'Hospital: . There are many other ways - I'll save them for the pleasure of others.
Last edited by mr fantastic; May 9th 2008 at 10:58 PM. Reason: Added last line
Now we know that converges because it's always greater than zero and the limit of is 1. So the limit is in indeterminate form . We can use L'hopital later. Define . Now it's . Use L'hopital:
Originally Posted by Chris L T521 I already evaluated the limit, and got the answer, but I want to see if you guys can come up with an easier way to solve it : . For once L'hopitals's rule is appropriate: (Note the integral in the expression on the right goes to infinity as does as the integrand goes to for large ) So RonL
Originally Posted by CaptainBlack For once L'hopitals's rule is appropriate: (Note the integral in the expression on the right goes to infinity as the integrand goes to for large ) So RonL The l'Hospital's busy tonight By the way, CaptainB ..... "For once L'hopitals's rule is appropriate" .... solid gold!
Originally Posted by mr fantastic The l'Hospital's busy tonight By the way, CaptainB ..... "For once L'hopitals's rule is appropriate" .... solid gold! Careless of me, I hadn't noticed the other replies I take that back, just looked at the cached version of the question, no replies when I started typing, so just slow typing on my or other's part RonL
Originally Posted by CaptainBlack Careless of me, I hadn't noticed the other replies RonL Well, if you hadn't replied, I wouldn't be wiping away tears of laughter right now.
Let's make this more generally: If
Originally Posted by Krizalid Let's make this more generally: If It is not as general as you wrote it. You need to satisfy some conditions. "If is a continous function and exists and is , and if then ."
Here is a solution without L'Hopital's rule. Note that, for all . Thus, . This gives us, . Dividing by gives, . Since it follows by the Squeeze theorem that .
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