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.)
$$
\lim _{x \rightarrow 2^{-}} f(x)=\infty, \lim _{x \rightarrow 2^{+}} f(x)=-\infty, \text { and } f(2)=1
$$

Key Concepts

One-Sided Limits

One-sided limits describe the behavior of a function as the input approaches a particular point from one direction only—either from the left (denoted as x ? c?) or the right (denoted as x ? c?). They are essential when a function behaves differently on each side of a point, as they allow us to analyze and describe the distinct behaviors of the function near that point, even when a two-sided limit may not exist.

Vertical Asymptote

A vertical asymptote occurs when the values of a function increase or decrease without bound as the independent variable approaches a particular value. This is typically observed when the function's expression leads to division by zero or another form of unbounded behavior at some point, causing the function to 'blow up' to infinity or negative infinity. Vertical asymptotes are key in understanding the behavior of graphs near points of discontinuity.

Discontinuities and Removable vs. Non-Removable Discontinuities

Discontinuities in a function occur when the function is not continuous at a specific point. A removable discontinuity is one where the function is not defined, or defined differently than the limit, at that point even though the limit exists from both sides. In contrast, non-removable discontinuities (such as those associated with vertical asymptotes) occur when the one-sided limits are infinite or do not converge to the same value, making it impossible to redefine the function at that point to ‘fill the gap’ in its behavior.

Graphing Functions with Discontinuous Behavior

Graphing functions that exhibit discontinuities involves plotting the behavior dictated by the limits and the function’s values at specific points. When a function has differing one-sided limits (for example, one side approaching infinity and the other negative infinity) but also has a defined value at the discontinuity, this extra information helps in constructing a graph that accurately reflects the dramatic differences in the behavior near the point of discontinuity. Such graphs must show the vertical asymptote and note the defined value, even if it does not match the behavior suggested by the asymptotic limits.

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