# Limits of 2 variables

1. Mar 31, 2008

### MAins

What is:
lim(x,y)->(0,0) of (x^2)(y) / ((x^4) + (y^2)) ?

When I take x_n = 0, y_n = 1/n, lim=0
and x_n = 1/n, y_n = 0, lim=0
and x_n = y_n = 1/n, lim=0
All three limits are zero, yet other people I've asked say the limit doesn't
exist. Am I right, or am I doing something wrong here? Thanks.

2. Mar 31, 2008

### rocomath

$$\lim_{(x,y) \rightarrow (0,0)}\frac{x^2y}{x^4+y^2}$$

$$x=0$$
$$\lim_{(x,y) \rightarrow (0,0)}\frac{0\cdot y}{0+y^2}=0$$

$$y=0$$
$$\lim_{(x,y) \rightarrow (0,0)}\frac{x^2\cdot 0}{x^4+0}=0$$

$$y=x^2$$
$$\lim_{(x,x^2) \rightarrow (0,0)}\frac{x^4}{2x^4}=\frac 1 2$$

Aim to make your powers the same, use $$y=x^2$$ or $$x=y^2$$.

General tests:

$$x=y=0$$
$$y=x$$
$$x=y^n$$
$$y=x^n$$

Last edited: Mar 31, 2008