limits of more than 1 variable?

Question

how do i solve limits that involve more than 1 variable eg limit of (x,y) ->(0,0) of (2x^2 * y) / (x^3 +4y^3)

Answer 1

A useful hint: if the limit approaches the origin and you have a quotient function, then if the degree of the numerator is less than or equal to the degree of the denominator, it is highly probable that the limit does not exist... (i dont know if there are counterexamples for this yet...)

Now, the degree of the numerator is 3, the degree of the denominator is also 3. Then i will show that the limit does not exist:

Take two curves P1 & P2 passing through the origin:

Let P1: y = 0
the limit becomes: limit [x→0] 0/x^3 = 0

Now, let P2: y=x
the limit becomes: limit [x→0] 2x^3/(x^3+4x^3) = 2/5

Theorem:
the limit of a multivariate function is unique, thus in any curve passing through the point, the limit value must all be the same.

Thus the limit does not exist.

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Answer 2

It is a very tricky problem.

1. You must do the limits one at a time, and the results must not depend on the order.

2. You can compound the two limits in one (e.g. y=t, x=t, lim t->0; or y=t^2, x=-2t, lim t->0; or y=1/t, x=1/t, lim t->infinity; ...) and the limit must be the same as before in any case

3. otherwise the limit does not exist.

4. this is related to the problem of analicity and complex field functions

Answer 3

It´s a good method to pass to polar coordinates
x=r cos t and y = r sin t
so f(x,y) = 2r^3 cos^2t*sin t / r^3(cos^3 t+4sin^3 t)=
2cos^2(t)*sin(t) / (cos^3 t+4 sin^3(t))
so f(x,y) as both ==> 0,0 depends on the angle t.
So the limit does not exist