Sketch the graphs of functions that have the given limits...

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 2^{-}} f(x)=3, \lim _{x \rightarrow 2^{+}} f(x)=3, \text { and } f(2)=0
$$

Key Concepts

Limit

A limit describes the behavior of a function as the input approaches a particular value, even if the function is not defined at that value. Here, the limit as x approaches the point is stated for both directions, indicating that the values of the function get arbitrarily close to a specific number as x nears the point from either side.

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a point from one specific side—either from the left or the right. In this context, both the left-hand limit and the right-hand limit are given, and they are equal to the same value, indicating that the function behaves similarly on both sides of the point.

Removable Discontinuity

A removable discontinuity occurs when a function has a limit at a point, but the function's value at that point either is not defined or does not equal the limit. In this problem, the limit is 3 but the function value f(2) is 0, creating a removable discontinuity that could be 'fixed' by redefining the function value to match the limit.

Graphing Functions with Discontinuities

Sketching the graph of a function that includes a removable discontinuity involves drawing a continuous curve approaching a certain value at the point of discontinuity and then indicating the differing function value at that point, often with an open circle at the limiting value and a filled dot at the actual function value. This visual representation helps to clearly convey the difference between the limit and the function's defined output.

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