1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Limit of two-variable function

  1. Oct 14, 2005 #1
    Hi all,

    suppose I want to get this:

    \lim_{[x,y] \rightarrow [0,0]} (x^2+y^2)^{xy}

    Here's how I approached:

    \lim_{[x,y] \rightarrow [0,0]} (x^2+y^2)^{xy} = \lim_{[x,y] \rightarrow [0,0]} \exp^{xy \log (x^2+y^2)}

    \lim_{[x,y] \rightarrow [0,0]} xy \log (x^2 + y^2) = (x^2 + y^2) \log (x^2 + y^2) \frac{xy}{x^2 + y^2} \rightarrow 0

    Because the last fraction is bounded and the part before it goes to 0 (I hope).

    But that's the problem, I don't know how to prove

    \lim_{t \rightarrow 0+} t\ \log t = 0

    Thank you for help.
  2. jcsd
  3. Oct 14, 2005 #2


    User Avatar
    Homework Helper

    You can use L'Hopital rule to prove that. Try rearrange the equation to:
    [tex]\lim_{t \rightarrow 0 ^ +} t \log{t} = \lim_{t \rightarrow 0 ^ +} \frac{\log{t}}{\frac{1}{t}}[/tex]. Now it's in form [tex]\frac{\infty}{\infty}[/tex]. Can you go from here?
    Viet Dao,
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook