Help limit problem multivariable

[yaho8888]

Homework Statement

lim (x^2+y^2)/((root(x^2+y^2+1) - 1)
(x,y)-->(0,0)
what is the limit

Homework Equations

none

The Attempt at a Solution

<br /> \lim_{(x,y) \to (0,0)} \frac {x^2 + y^2}{\sqrt{x^2 + y^2 + 1} - 1} = \lim_{r \to 0} \frac {r^2}{\sqrt{r^2 + 1} - 1}<br />

I got this far the answer are 2 but i don't know how it is 2.

 

[yaho8888]

how come no one is helping?

 

[HallsofIvy]

Perhaps because people are not sitting around with nothing to do but answer your questions! You waited a whole 29 minutes? Patience, grasshopper.

Switching to polar coordinates is a very good idea. That way, (x,y) going to (0,0) is the same as the single variable, r, going to 0. As long as the result is independent of the angle \theta, that is the limit. Now, the difficulty is that when you substitute r= 0 in the fraction, you get "0/0". Do you remember any methods from Calculus I for doing that? Perhaps L'Hopital's rule? Or maybe "rationalizing the denominator" by multiplying both numerator and denominator by \sqrt{r^2+ 1}+ 1

 

[yaho8888]

thanks for the help. I got it! (one more thing, I am a grasshopper!) :)