True or False. Explain.
The graph of any function can have at most two horizontal asymptotes.
True, if more then not a function
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False.
To determine the number of horizontal asymptotes of a function, we need to consider the behavior of the function as x approaches positive infinity and negative infinity.
There are three possibilities for the number of horizontal asymptotes:
1. The graph can have no horizontal asymptotes.
2. The graph can have one horizontal asymptote.
3. The graph can have two horizontal asymptotes.
The number of horizontal asymptotes depends on the behavior of the function at the extremes of its domain.
To find the horizontal asymptotes, we can use the limit definition.
1. If the limit as x approaches positive infinity (or negative infinity) of the function is a finite number, then there is a horizontal asymptote at that value.
2. If the limit as x approaches positive infinity and the limit as x approaches negative infinity both exist and are finite, then there are two horizontal asymptotes.
3. If the limit as x approaches positive infinity or negative infinity does not exist or is infinite, then there are no horizontal asymptotes.
Therefore, the graph of any function can have no horizontal asymptotes, one horizontal asymptote, or two horizontal asymptotes.