(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

a) Prove that;

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

b) Prove that if [tex]\lim_{(x,y)\rightarrow(0,0)} f(x,y) = L_1[/tex] and [tex]\lim_{(x,y)\rightarrow(0,0)} f(x,y) = L_2[/tex], then [tex]L_1=L_2[/tex]

c) Using the statement proven in 5b, prove that

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

Does NOT exist.

2. The attempt at a solution

a)

[tex]f(0,y) = \frac{0}{y^2} = 0[/tex]

[tex]f(x,0) = \frac{0}{x^2} = 0[/tex]

From those 2 directions, the limit is the same, so;

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

b)

I have no idea how to do that, it seems to evident !

c)

Does NOT exist ? But...

[tex]f(0,y) = \frac{0}{y^2} = 0[/tex]

[tex]f(x,0) = \frac{0}{x^2} = 0[/tex]

It's exactly the same thing as in a), the limit DOES exist and it is;

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Limits (another error in the question?)

**Physics Forums | Science Articles, Homework Help, Discussion**