example of when a limit does not exist for a function of two variables 1

The function is defined as f(x, y) = x(ax²)²/(x²+(ax²)⁴). If we move towards the origin along the line y=x, we get the limit lim(x, y) → (0, 0) f(x, y) = 0. However, if we move towards the origin along the line y=mx, where m is not equal to 1, we get the limit lim(x, y) → (0, 0) f(x, y) = 0. This means that the limit of the function does not exist at the origin. The reason for this is that the function has a sharp turn at the origin. As we move towards the origin along the line y=x, the function approaches 0 smoothly. However, as we move towards the origin along the line y=mx, the function approaches 0 in a sharp way. This is because the denominator of the function approaches 0 much faster than the numerator along the line y=mx. This is an example of a function whose limit does not exist because it is not continuous at the origin. A function is continuous at a point if and only if the limit of the function at that point exists. In this case, the limit of the function does not exist because the function has a sharp turn at the origin.