It's been a while since I've evaluated limits, and I'm beginning to forget some of the techniques. A problem came up in physics which involved evaluating a limit of this particular form.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

[tex]\lim_{x \to \infty} \left( \frac{x}{\sqrt{x^2+y^2}} \right)[/tex]

2. Relevant equations

L'hopital's rule would be my first guess at the proper approach.

3. The attempt at a solution

I know the limit will evaluate to be a 1. This could also be inferred from the fact that x is in the same degree in both the numerator and the denominator. But I tried to do it a bit more rigorously with l'Hopital's rule. The problem is that the expression inside the limit becomes circular with successive derivatives. That is, if we let [itex]f(x) = x[/itex] and [itex]g(x)=\sqrt{x^2+y^2}[/itex], we will find that:

[tex]\frac{f'(x)}{g'(x)}=\frac{g(x)}{f(x)}[/tex]

This will bring us no closer to finding the limit. Take the derivative of the numerator and the denominator again to find:

[tex]\frac{f''(x)}{g''(x)}=\frac{f(x)}{g(x)}[/tex]

Hey, we're back! L'Hopital took us for a spin and brought us back to the starting point.

So my questions:

1. What is the proper approach to evaluate the limit?

2. For the sake of curiosity, are there any interesting observations to be made about the situation? Perhaps a name for the circular nature of the problem?

Thanks in advance for any help.

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# Evaluating a limit with l'hopital's rule

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