How many horizontal asymptotes can the graph of $y = f (x)$ $y=f(x)$ have?

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How many horizontal asymptotes can the graph of $y = f (x)$ $y=f(x)$ have?

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Answer 1 · 2014-09-21T04:02:56

The answer is 0, 1, or 2.

You have to check the end behavior at $\pm \infty$ $+-oo$ , because they don't have to match.

If the growth rate of the numerator is faster than that of the denominator, you won't have a horizontal asymptote. For example, $f (x) = x^{2}$ $f(x)=x^2$ , it is implied that the denominator is $1$ $1$ .

If the growth rate of the denominator is faster than that of the numerator, then the horizontal asymptote is $y = 0$ $y=0$ . For example, $f (x) = \frac{1}{x}$ $f(x)=1/x$ .

If the growth rate of the numerator and denominator differ by a constant, $c$ $c$ , then the horizontal asymptote is $y = c$ $y=c$ . Here is a graphical example with 2 horizontal asymptotes $y = - 1$ $y=-1$ and $y = 1$ $y=1$ :
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How many horizontal asymptotes can the graph of $y = f (x)$ $y=f(x)$ have?

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