(a) Can the graph of y=f(x) intersect a horizontal...

(a) Can the graph of $y=f(x)$ intersect a horizontal
asymptote? If so, how many times? Illustrate by
sketching graphs.
(b) How many horizontal asymptotes can the graph of $y=f(x)$ have? Sketch graphs to illustrate the possibilities.

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Key Concepts

Intersection of Graph with Horizontal Asymptote

Although a horizontal asymptote describes the end behavior of a function, the graph of the function can intersect its horizontal asymptote at finite points. The existence of a horizontal asymptote only guarantees that the function approaches the line as x becomes very large or very small, but does not preclude the possibility of crossing the asymptote at other points.

Limits at Infinity

Limits at infinity are used to analyze the behavior of functions as x approaches positive or negative infinity. They are crucial for determining horizontal asymptotes, as a function f(x) has a horizontal asymptote y = L if the limit of f(x) is L as x approaches infinity or negative infinity. Understanding these limits provides insights into the end behavior of functions beyond local variations.

Horizontal Asymptote

A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as x tends to positive or negative infinity. It is determined by the limit of the function as x approaches infinity or negative infinity. This concept is critical in understanding the long-term behavior of functions and is a key element in studying end behavior in calculus.

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