My Math Forum uni level convergence/limit question

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 December 3rd, 2013, 01:44 AM #1 Newbie   Joined: Dec 2013 Posts: 4 Thanks: 0 uni level convergence/limit question I need to be able to explain the solution to the limit: $\lim\limits_{x \to \infty}\dfrac{\sqrt{x^{2}+1}-x}{sin(1/x)}$. However, I am unable to solve it. I have tried using l'Hopital's rule but that seems to make it too complicated, and the squeeze theorem makes it too simplified and I can't get to the desired answer which is 1/2 (I eventually looked up the graph in the hope that it would make it easier to solve). Any help would be great.
 December 3rd, 2013, 10:58 AM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,312 Thanks: 115 Re: uni level convergence/limit question Have you tried hitting the numerator with the Binomial Theorem?
December 3rd, 2013, 12:34 PM   #3
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Re: uni level convergence/limit question

Quote:
 Originally Posted by RedPanda624 I need to be able to explain the solution to the limit: $\lim\limits_{x \to \infty}\dfrac{\sqrt{x^{2}+1}-x}{sin(1/x)}$. However, I am unable to solve it. I have tried using l'Hopital's rule but that seems to make it too complicated, and the squeeze theorem makes it too simplified and I can't get to the desired answer which is 1/2 (I eventually looked up the graph in the hope that it would make it easier to solve). Any help would be great.
Let y = 1/x and look at the problem as y -> 0. You will find it a lot easier (L'Hopital).
Hint: step2 - multiply by y/y, before L'Hopital.

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