Limit of (x²+y²)/(√(x²+y²+1)−1) as (x,y)→(0,0)

[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

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

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!) :)