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lets_resonate
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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.
[tex]\lim_{x \to \infty} \left( \frac{x}{\sqrt{x^2+y^2}} \right)[/tex]
L'hopital's rule would be my first guess at the proper approach.
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.
Homework Statement
[tex]\lim_{x \to \infty} \left( \frac{x}{\sqrt{x^2+y^2}} \right)[/tex]
Homework Equations
L'hopital's rule would be my first guess at the proper approach.
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.