Continuity for a function with two independent variables

Find the limit if it exists.

Example:

lim x³y / ( 2x⁴ + 3 y⁴ )

(x,y) → (1,1)

The function f(x,y) = x³y / ( 2x⁴ + 3 y² ) is continuous in the plane so

f(1,1) = 1/(2 + 3) = 1/5

Example:

lim x³y / ( 2x⁴ + 3 y⁴ )

(x,y) → (0,0)

Although f(x,y) = x³y / ( 2x⁴ + 3 y⁴ ) is a rational function it is undefined at (0,0).

So substituting in x = 0 and y = 0 does not apply. Instead:

Investigate the limit along y = 0. i.e. the x-axis

For x ≠ 0 lim x³y / ( 2x⁴ + 3 y² ) = 0 / 2x⁴ = 0

(0,y) → (0,y)

Investigate along y = x

lim x⁴ / ( 2x⁴ + 3 x⁴ ) = 1 / 5

(0,y) → (0,y)

Since the limits for these two directions differ, the limit does not exist.

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