Question

check whether the limit of the function f (x,y)=3x^3y/x^6+2y ^2 exists as (x,y) -(0,0)

Answer 1

Let y=x.

$\frac{3x^3y}{x^6+2y^2} = \frac{3x^4}{x^6+2x^2} = \frac{3x^2}{x^4+2} \underset{x\to0}{\to} 0$

Let y=x^3.

$\frac{3x^3y}{x^6+2y^2} = \frac{3x^6}{x^6+2x^6} = \frac{3}{3} \underset{x\to0}{\to} 1$

Since the limits in two special cases are different, the limit of the given function does not exist as (x,y) tends to (0,0).