One way to look at it is consider what the expression approaches at
(x,y)=(x,0).
The limit approaches 1 at all points on the x-axis except the origin:
\(\displaystyle \L\\\frac{x^{4}}{x^{4}+0}\).
Likewise, the values approach 0 as (x,y)=(0,y) approaches the origin
along the y-axis:
\(\displaystyle \L\\\frac{0}{0+y^{2}}\)
Since the values approach different numbers as (x,y) approaches the
origin, the limit at the origin does not exist.
If f(x,y) approaches different numbers as (x,y) approaches (a,b) along
different curves, then \(\displaystyle \L\\\lim_{(x,y)\to\(0,0)}f(x,y)\) does not
exist.
