2. LIMITS AND CONTINUITY
61
2x2y
Example. Find lim
.
(x,y)!(0,0) x4 + y 2
For x = 0 (approaching the origin along the y-axis),
2x2y
0
=
= 0 ! 0 as y ! 0.
x4 + y 2 y 2
For y = 0 (approaching the origin along the x-axis),
2x2y
0
=
= 0 ! 0 as x ! 0.
x4 + y 2 x4
But, along the parabola y = x2,
2x2y
2x4
2x4
=
=
= 1 ! 1 as (x, y) ! (0, 0).
x4 + y 2 x4 + x4 2x4
2x2y
Since we now have di↵erent limits along two paths, lim
does not
(x,y)!(0,0) x4 + y 2
exist (DNE).
Showing that a limit exists
Theorem (2.1). Suppose |f (x, y) L|  g(x, y) for all (x, y) in the
interior of some circle centered at (a, b), except possibly at (a, b).
If
lim
(x,y)!(a,b)
g(x, y) = 0, then
lim
(x,y)!(a,b)
f (x, y) = L