For the function g(x) graphed as follows, approximate each...

For the function $g(x)$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow 1^{-}} g(x), \lim _{x \rightarrow 1^{+}} g(x), \lim _{x \rightarrow 1} g(x), \text { and } g(1) .
$$

Key Concepts

Difference Between Limits and Function Values

The value of a limit describes what a function approaches as the input nears a particular point, whereas the function value is the actual output at that point. This distinction is especially important in the context of discontinuities, where the limit and the function value may differ.

Continuity and Discontinuity

Continuity at a point means the function’s value at that point matches the limit as the variable approaches that point, while discontinuity occurs when there is a break or jump, or when the limit does not equal the function’s value. Analyzing limits helps in understanding where and why these breaks occur.

One-Sided Limits

One-sided limits describe the behavior of a function as its input approaches a particular value from one specific direction—either from the left (negative side) or from the right (positive side). They are crucial when a function behaves differently on each side of a point, especially near discontinuities or jump points.

Two-Sided Limits

Two-sided limits consider the behavior of a function as the input approaches a specific point from both the left and the right. For a two-sided limit to exist, the one-sided limits from both directions must be equal, providing a single, unified limit value.

Graphical Analysis of Functions

Graphical analysis involves interpreting the visual representation of a function to determine its behavior, including estimating limit values. By examining the graph, one can observe how the function behaves near points of interest and identify discontinuities or other features.

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