Understanding the Two-Sided Limit in Mathematics | Definition, Conditions, and Importance

When does the two sided limit exist

The two-sided limit, also known as the limit from both sides or the limit at a point, exists when the limit of a function exists as the input approaches a certain value from both the left and right sides

The two-sided limit, also known as the limit from both sides or the limit at a point, exists when the limit of a function exists as the input approaches a certain value from both the left and right sides.

Formally, we say that the two-sided limit of a function f(x) as x approaches a value c exists if and only if the following two conditions are satisfied:

1. The one-sided limit from the left side, denoted as f(x-) or lim(x→c-) f(x), exists. This means that the limit of f(x) as x approaches c from the left side exists.

2. The one-sided limit from the right side, denoted as f(x+) or lim(x→c+) f(x), exists. This means that the limit of f(x) as x approaches c from the right side exists.

If both conditions are met, we can say that the two-sided limit of f(x) as x approaches c exists and is typically denoted as lim(x→c) f(x) or simply lim f(x).

It is important to note that in order for the two-sided limit to exist, the one-sided limits from both the left and right sides must be finite and equal. If they are equal, the two-sided limit will also be equal to that value. However, if the one-sided limits are not equal, then the two-sided limit does not exist.

In summary, the two-sided limit exists if and only if the one-sided limits from both sides exist and are equal.

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