In these exercises, make reasonable assumptions about the...

In these exercises, make reasonable assumptions about the graph of the indicated function outside of the region depicted. For the function $g$ graphed in the accompanying figure, find $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 1^{-}} g(x)} & {\text { (b) } \lim _{x \rightarrow 1^{+}} g(x)} \\ {\text { (c) } \lim _{x \rightarrow 1} g(x)} & {\text { (d) } g(1)}\end{array} $$

In these exercises, make reasonable assumptions about the graph of the indicated function outside of the region depicted.
For the function $g$ graphed in the accompanying figure, find
$$
\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 1^{-}} g(x)} & {\text { (b) } \lim _{x \rightarrow 1^{+}} g(x)} \\ {\text { (c) } \lim _{x \rightarrow 1} g(x)} & {\text { (d) } g(1)}\end{array}
$$

Key Concepts

Graphical Interpretation of Functions

Graphical interpretation involves analyzing the visual representation of a function to understand its behavior, including limits, continuity, and points of discontinuity. This skill is crucial when making inferences about function behavior based on a graph, particularly when extrapolating or making reasonable assumptions about regions not explicitly shown.

Continuity

Continuity of a function at a point involves the condition that the limit of the function as the input approaches the point equals the function’s actual value at that point. If this condition is not met, the function is said to be discontinuous, which can be due to jumps, holes, or asymptotic behavior.

One-Sided Limits

One-sided limits refer to the behavior of a function as the input approaches a particular point from one side only — either from the left (values less than the point) or from the right (values greater than the point). This concept is essential for understanding the behavior of functions near points where the function might have different values or behaviors on either side.

Limit of a Function at a Point

The limit of a function at a point describes what value the function approaches as the input values get arbitrarily close to that point, regardless of what the actual function value might be at that point. For the overall limit to exist, the left-hand and right-hand limits must both exist and be equal.

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