How And Why Limits Of Math Functions Fail To Exist: Oscillations, Asymptotes, Discontinuity, And Multiple Paths

limit of f(x,y) does not exist

if the function does not approach the same limit from every path

When the limit of a function f(x,y) does not exist, it means that as we approach a certain point (x0,y0) in the domain of the function, the values of f(x,y) become increasingly erratic or divergent. In other words, there are multiple possible paths along which the function approaches different values as we get close to (x0,y0), making it impossible to determine the behavior of the function at that point through direct evaluation.

Here are a few reasons why a limit might not exist:

1. Oscillations: The function oscillates or fluctuates as it approaches (x0,y0) along different paths, preventing it from converging to a single value.

2. Asymptotes: The function may have vertical or slanted asymptotes as we approach (x0,y0), causing it to diverge towards infinity or negative infinity.

3. Discontinuity: The function may have a discontinuity at (x0,y0), i.e., there is a jump or a hole in the graph of the function at that point.

4. Multiple paths: Different paths along which the function is approaching (x0,y0) give rise to different values. This happens when the function takes on different values in the surrounding neighborhoods of (x0,y0).

To summarize, when the limit of a function does not exist, it means that there is no unique value towards which the function is approaching as we get closer to a particular point in its domain.

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