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.

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Key Concepts

Graphical Analysis of Functions

Graphical analysis involves examining the graph of a function to deduce its behavior near specific points, particularly limits and continuity. It is a visual approach to verifying limits and identifying any discontinuities, which can include jumps, holes, or asymptotic behavior.

Continuity

Continuity of a function at a point means that the function not only has a limit at that point, but the function’s value matches this limit, resulting in no breaks, jumps, or holes in the graph. The discussion of continuity involves verifying that the left-hand limit, right-hand limit, and the function's value at that point are all equal.

Limit

A limit is a fundamental concept in calculus that describes the value that a function approaches as the input nears a particular point. It provides an understanding of the behavior of functions at points where they might not be explicitly defined or where direct evaluation is challenging.

One-Sided Limit

One-sided limits examine the behavior of a function as the input approaches a specified point from one side only—either from the left (denoted by x ? c?) or from the right (denoted by x ? c?). These limits are crucial for understanding functions that might have different behaviors on either side of a point.

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