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Relationship Between the Limit and One-Sided Limits

General Limit​

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

lim⁡x→af(x)=L\lim_{x \to a} f(x) = L

One-Sided Limits​

The one-sided limits are represented as:

lim⁡x→a+f(x)andlim⁡x→a−f(x)\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)f(x) as xx approaches aa exists and is equal to LL, then both one-sided limits must exist and both must be equal to LL as well.

    lim⁡x→af(x)=L⇒lim⁡x→a+f(x)=lim⁡x→a−f(x)=L\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→a+f(x)=lim⁡x→a−f(x)⇒lim⁡x→af(x)=L \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→a+f(x)≠lim⁡x→a−f(x)⇒lim⁡x→af(x) 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=ax = 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​