Limit fails to exist if the function…
1. Jumps2. Grows to large, increases or decreases with bound (+- infinity)3. Oscillates
There are several scenarios where the limit may fail to exist for a given function. Here are a few possibilities:
1. Oscillations: The function oscillates or fluctuates around the value of the limit as x approaches a certain value but doesn’t approach any specific value. In this case, the limit fails to exist. For example, consider the function f(x) = sin(1/x). As x approaches 0, f(x) oscillates between -1 and 1, and doesn’t approach any specific value. Therefore, the limit fails to exist at x = 0.
2. Jump Discontinuity: The function has a jump discontinuity at some value of x. In other words, the limit from the left and the limit from the right of that value of x are not equal. For example, consider the function g(x) = {x, if x < 0; and x+1, if x >= 0}. As x approaches 0 from the left, g(x) approaches 0, whereas as x approaches 0 from the right, g(x) approaches 1. Therefore, the limit of g(x) as x approaches 0 does not exist.
3. Infinite Limit: The function approaches infinity or negative infinity as x approaches a certain value. In this case, the limit fails to exist. For example, consider the function h(x) = 1/x. As x approaches 0 from the left or right, h(x) approaches negative infinity, whereas as x approaches infinity, h(x) approaches 0. Therefore, the limit of h(x) as x approaches 0 does not exist.
4. Unbounded Function: The function is unbounded as x approaches a certain value. In other words, the function grows infinitely large or becomes infinitely small as x approaches a certain value. For example, consider the function f(x) = e^(1/x). As x approaches 0 from the right, f(x) grows infinitely large, whereas as x approaches 0 from the left, f(x) approaches 0. Therefore, the limit of f(x) as x approaches 0 does not exist.
These are some of the scenarios in which the limit may fail to exist for a given function.
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