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

Graphical Interpretation of Limits

Graphical interpretation of limits involves analyzing the visual behavior of a function on its graph near a particular point. By observing the function's trend as it approaches a point from either side, one can infer the existence of one-sided limits and overall limits, as well as determine if any discontinuities, such as holes or jumps, are present in the function.

Two-Sided Limit

A two-sided limit exists at a point if and only if both the left-hand limit and the right-hand limit exist and are equal. This concept is foundational in understanding continuous behavior in functions, as the overall limit reflects the behavior of the function as the input approaches a specific value from all directions.

One-Sided Limits

One-sided limits refer to the behavior of a function as the variable approaches a specific value from one direction only—either from the left (denoted as x ? c?) or from the right (denoted as x ? c?). This concept is critical because sometimes the behavior of a function differs depending on the direction of the approach, and understanding this helps in analyzing discontinuities or abrupt changes in the function’s graph.

Continuity

Continuity at a point means that a function does not have any breaks, jumps, or holes at that particular point. A function is continuous at a point if the limit as the input approaches that point exists and equals the function’s value at that point. This concept links the ideas of limits and function evaluation, ensuring smooth behavior in mathematical models.

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