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Homework Help: Help limit problem multivariable!

  1. Feb 25, 2008 #1
    1. The problem statement, all variables and given/known data
    lim (x^2+y^2)/((root(x^2+y^2+1) - 1)
    what is the limit

    2. Relevant equations


    3. 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 dont know how it is 2.
  2. jcsd
  3. Feb 25, 2008 #2
    how come no one is helping?
  4. Feb 25, 2008 #3


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    Science Advisor

    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 [itex]\theta[/itex], 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 [itex]\sqrt{r^2+ 1}+ 1[/itex]
  5. Feb 25, 2008 #4
    thanks for the help. I got it! (one more thing, I am a grasshopper!) :)
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