Determine whether the statement is true or false. Explain...

Determine whether the statement is true or false. Explain your answer.
$$
\begin{array}{l}{\text { If } \lim _{x \rightarrow a} f(x) \text { exists, then so do } \lim _{x \rightarrow a^{-}} f(x) \text { and }} \\ {\lim _{x \rightarrow a^{+}} f(x) .}\end{array}
$$

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Key Concepts

Limit

A limit describes the value that a function approaches as the input approaches a specific point. This concept is foundational in calculus, where establishing the behavior of a function near a point—even if the function is not defined at that point—is crucial for analyzing continuity, differentiability, and the overall behavior of functions.

One-sided Limit

A one-sided limit considers the behavior of a function as the input approaches a specific point from one side only—either from the left (negative side) or the right (positive side). This concept helps in understanding the behavior of functions at points where there might be discontinuities or other irregularities that differ when approached from different directions.

Relationship between Two-sided and One-sided Limits

If a two-sided limit exists at a point, it means that as the input approaches that point from either the left or right side, the function approaches the same value. Consequently, the existence of the two-sided limit implies that both one-sided limits exist and are equal to the overall limit. This relationship is a key criterion in determining the continuity of a function at a given point.

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