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.)
$$
\lim _{x \rightarrow-\infty} f(x)=3 \text { and } \lim _{x \rightarrow \infty} f(x)=-\infty
$$

Key Concepts

End Behavior

End behavior describes how a function behaves as inputs approach extreme values, either positive or negative infinity. It is closely linked to limits at infinity and asymptotes, and determines the overall trend or direction of the graph in its far left and right regions.

Graph Sketching

Graph sketching involves drawing a visual representation of a function based on its mathematical properties, such as limits, asymptotes, and critical points. It is a key skill in mathematical analysis, helping to illustrate and understand how the function behaves across its domain, particularly its tendencies towards infinity or specific finite limits.

Limits at Infinity

Limits at infinity refer to the behavior of a function as the independent variable approaches very large positive or very negative values. This concept is used to describe how the function stabilizes, diverges, or oscillates as x moves towards positive or negative infinity, providing insight into the end behavior of the function.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that the graph of a function approaches as x tends to plus or minus infinity. When a function approaches a constant value as x goes to infinity or negative infinity, that constant is the y-value of the horizontal asymptote, which helps in understanding the function's long-term behavior.

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