In Exercises 1-6, use the graph to determine the limit, and...

In Exercises $1-6,$ use the graph to determine the limit, and discuss the continuity of the function. $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow c^{+}} f(x)} & {\text { (b) } \lim _{x \rightarrow c^{-}} f(x)} & {\text { (c) } \lim _{x \rightarrow c} f(x)}\end{array} $$ Graph cannot copy.

In Exercises $1-6,$ use the graph to determine the limit, and discuss the continuity of the function.
$$
\begin{array}{lll}{\text { (a) } \lim _{x \rightarrow c^{+}} f(x)} & {\text { (b) } \lim _{x \rightarrow c^{-}} f(x)} & {\text { (c) } \lim _{x \rightarrow c} f(x)}\end{array}
$$
Graph cannot copy.

Key Concepts

One-Sided Limit

A one-sided limit examines the behavior of a function as the input values approach a specific point from one direction only—either from values less than the point (left-hand limit) or values greater than the point (right-hand limit). This concept is crucial when a function behaves differently on either side of the point, as it helps identify discontinuities and other subtle properties of the function.

Two-Sided Limit

The two-sided limit considers the behavior of the function as the input approaches a particular value from both directions simultaneously. For a two-sided limit to exist, the left-hand and right-hand limits must both exist and be equal. This concept underpins much of limit theory and is essential in defining the overall behavior of functions near points of interest.

Continuity

Continuity at a point means that a function's value at that point seamlessly matches the limit of the function as the input approaches that point. In other words, a function is continuous at a point if its left-hand limit, right-hand limit, and the function’s value are all equal. Graphically, a continuous function can be drawn without lifting the pencil from the paper, indicating no breaks, jumps, or holes in the graph.

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