Find the limit?

Question

Find lim as (x, y)--->(0, 0) for (8x^2*y^2)/(x^4+y^4).

lim as (0, y)--->(0, 0) =0/y^4=0

lim as (x, 0)--->(0, 0) =0/x^4=0

The limit is 0, right?

But doesn't exist is the answer for this problem. Did I do something wrong?

Answer 1

when you take lim as (0, y)--->(0, 0), you're approaching (0,0) along the y-axis.

Likewise, when you take lim as (x, 0)--->(0, 0), you're approaching (0,0) along the x-axis.

However, if you try approaching (0,0) along the line y=x, you have lim as (x,x)--->(0,0) = (8x^2*x^2)/(x^4+x^4) = 8x^4/2x^4 = 4.

Thus the limit is path dependent and does not exist. You can see this visually by looking at the graph of the equation in 3 dimensions here:

http://www.wolframalpha.com/input/?i=plot+%288x^2*y^2%29%2F%28x^4%2By^4%29

You can clearly see the limit is 0 along the two axes and 4 along the line y=x.

Answer 2

Incorrect. If y = x, then the limit = 4. If y = 2x, then the limit = 32/17.

Since different paths lead to different limits, the limit does not exist.