Exploring the Scenarios Where the Limit of a Function May Not Exist | Jump Discontinuity, Infinite Discontinuity, Oscillating Behavior, and Undefined Function

When does the lim f(x) x->c not exist?

The limit of a function f(x) as x approaches a certain value c may not exist in the following scenarios:

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The limit of a function f(x) as x approaches a certain value c may not exist in the following scenarios:

1. Jump Discontinuity: If f(x) has a jump discontinuity at x = c, it means that the function has a sudden jump at that point where the left and right limits do not match. In this case, the limit as x approaches c does not exist.

2. Infinite Discontinuity: If f(x) has an infinite discontinuity at x = c, it means that either the left or right limit (or both) approach positive or negative infinity. When the limit tends towards infinity or negative infinity, it is considered undefined.

3. Oscillating Behavior: If f(x) oscillates infinitely as x gets closer to c, with no particular pattern or convergence, then the limit may not exist. This is also known as oscillatory behavior, where the function does not approach a specific value.

4. Undefined Function: If f(x) is not defined at x = c, then the limit as x approaches c does not exist. This can occur when the function has a hole or point of non-existence at that particular point.

It is important to note that a limit exists if and only if the left-hand and right-hand limits are equal. If they are unequal, the limit does not exist.

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