By considering different paths of approach, show that the limit does not exist.

Question

lim (x,y)→(0,0) (xy*cosy)/(3x^2 + y^2 )

Answer 1

If two different paths to a point P(a, b) produce two different limiting values for f, then the limit of f(x, y) as (x, y) approaches (a, b) does not exist.

The function f(x,y) = (xy cos y)/(3x²+y²) is defined everywhere except at (0,0). To show that the function has no limit, we need to find two different paths to (0,0) that have different limiting values.

If approaching (0,0) from quadrant 1, the limit is positive. At (1,1), we get (1•1•cos1)/4 ≈ 1/4

If approaching (0,0) from quadrant 2, we can see that the limit is negative. At (-1,1), we get (-1)(1)(cos1)/4 ≈-1/4.

Hence, two different paths to (0,0) produce two different limiting values for the function.

Thus, the limit of f(x,y) as (x,y) approaches (0,0) does not exist.