Relationship Between the Limit and One-Sided Limits

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General Limit

For a function $f(x)$, the general limit at $x = a$ is represented as:

$$\lim_{x \to a} f(x) = L$$

One-Sided Limits

The one-sided limits are represented as:

$$\lim_{x \to a^+} f(x) \quad \text{and} \quad \lim_{x \to a^-} f(x)$$

Relationship

The relationship between the general limit and the one-sided limits is as follows:

• If the limit of $f(x)$ as $x$ approaches $a$ exists and is equal to $L$, then both one-sided limits must exist and both must be equal to $L$ as well.

  $$\lim_{x \to a} f(x) = L \Rightarrow \lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x) = L$$

• Conversely, if both one-sided limits exist and are equal to each other, then the general limit exists and is equal to this common value.

  $$\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x) \Rightarrow \lim_{x \to a} f(x) = L$$

• If the one-sided limits are not equal, then the general limit does not exist.

  $$\lim_{x \to a^+} f(x) \neq \lim_{x \to a^-} f(x) \Rightarrow \lim_{x \to a} f(x) \text{ Does Not Exist}$$

In summary, for the limit at $x = a$ to exist, the one-sided limits must exist and be equal. If the one-sided limits differ, the general limit at that point is undefined.

References

• calculus_cheat_sheet_limits | tutorial.math.lamar.edu