Limits: When A Function Fails To Reach A Definite Value At A Point

When does a limit not exist (one sided limit)

If two limits on the same graph arelim (x→ 0^+) f(x) = 1lim (x→ 0^-) f(x) = 0 Then the limit lim (x→ 0) f(x) = 0 does not exist because x approaching 0 from the left and the right create different numbers. Also, this type of limit is called a one sided limitAlso, I think lim (x→ a) f(x) = L is basically saying that if x approached a from the left OR from the right it would still create L. Therefore (x→ a) f(x) = L can only exist if lim (x→ a^-) f(x) = L. and lim (x→ 1^+) f(x) = L

A limit does not exist for a function f(x) at a point c in one of the following situations:

1. The limit from the left and the limit from the right do not agree, that is, the left-hand limit and the right-hand limit are not equal. In this case, the two one-sided limits do not exist at c, and therefore the limit does not exist.

2. The function oscillates between two or more values as x approaches c from either the left or right. This situation is known as oscillation, and the limit is said to not exist.

3. The function approaches infinity or minus infinity as x approaches c from either the right or the left. In this case, the limit does not exist because it is not a real number.

4. The function approaches a vertical asymptote at c from the left or the right.

5. The function has a jump discontinuity at c meaning the function jumps from one value to another at the point c.

It is important to note that for a limit to exist, both the left-hand and right-hand limits have to exist and be equal to each other.

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