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.)
$$
\lim _{x \rightarrow 3^{-}} f(x)=2, \lim _{x \rightarrow 3^{+}} f(x)=2, \text { but } f(3) \text { does not exist }
$$

Key Concepts

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a particular point from one side only—either from the left (negative direction) or from the right (positive direction). They are essential because they allow us to analyze how a function behaves near a point even if the function’s overall limit does not exist or the function is not fully defined at that point.

Limit at a Point

The limit of a function at a point examines the value that the function’s outputs approach as the input nears that point from both sides. When both one-sided limits exist and are equal, a two-sided limit exists at that point. This concept is fundamental in calculus for understanding continuity and the behavior of functions near critical points.

Removable Discontinuity

A removable discontinuity occurs when a function’s limit exists at a point, but the function is either not defined or is misdefined at that point. This kind of discontinuity can be ‘fixed’ or ‘removed’ by appropriately defining or redefining the function at that specific point, emphasizing the difference between the limit of a function and the function’s actual value.

Graphical Representation of Discontinuities

Graphing functions with discontinuities requires indicating that while the function approaches a certain value as it nears a point from either side, there is a break or gap at that point. This is often depicted by drawing an open circle or a gap on the graph, highlighting that the limit exists but the value at the point is not defined.

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