Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Evaluating a limit of a function of two variables

Tags:
  1. Jul 9, 2015 #1
    I want to evaluate [itex]\displaystyle\lim_{(x,y)\to(-1,0)}\frac{y^4(x+1)}{|x+1|^3+2|y|^3}[/itex]

    With some help, I was able to prove that the limit is 0, using Hölder's inequality. Like this:

    [tex]\left(|x+1|^3\right)^{1/5}\left(\frac{1}{2}|y|^3\right)^{4/5}\leq\frac{1}{5}|x+1|^3+\frac{4}{5}\frac{1}{2}|y|^3[/tex]

    Raising to the [itex]5/3[/itex] power and cancelling we get:

    [tex]|x+1|\left(\frac{1}{2}\right)^{4/3}y^4\leq\left(\frac{1}{5}\right)^{5/3}\left(|x+1|^3+2|y|^3\right)^{5/3}\\
    \frac{|x+1|y^4}{|x+1|^3+2|y|^3}\leq\sqrt[3]{\frac{16}{3125}}\left(|x+1|^3+2|y|^3\right)^{2/3}[/tex]

    But I wonder if there are any other ways to prove it. Does anyone have other ideas?

    Thanks for any input.
     
  2. jcsd
  3. Jul 10, 2015 #2
    We used to use the word DANG to remind students of the four ways to approach limits:

    Direct substitution
    Algebraic manipulation
    Numerical approximation
    Graphing
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Evaluating a limit of a function of two variables
Loading...