- #1

- 48

- 0

## Homework Statement

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]