This week we learned about horizontal asymptotes. First of all, to find a horizontal asymptote, you take the limit as x approaches infinity. Your answer should be set up as… y = #.
Second of all, how to find the limit is quite simple:
If the degree of the top is greater than the degree of the bottom, the limit is infinity.
If the degree of the bottom is greater than the degree of the top, the limit is 0.
If the degree of the top and bottom are the same, then you take the coefficient of the variable raised to the greatest degree of the top over the coefficient of the variable raised to the greatest degree of the bottom. And that's your limit.
Now, if you limit is a number, then that is also your horizontal asymptote. But if you limit is infinity, then you have no horizontal asymptotes.
Ex. 1) Find the horizontal asymptote(s) of f(x) = (x + 4) / (2x^2 + 5).
The degree of the bottom is greater than the top, so the limit is 0.
Therefore, y = 0 is a horizontal asymptote.
Ex. 2) Find the horizontal asymptote(s) of f(x) = (x^2 + 2x + 4) / x.
The degree of the top is greater than the bottom, so the limit is infinity.
Therefore, there are not horizontal asymptotes.
Ex. 3) Find the horizontal asymptote(s) of f(x) = (3x + 4x^2 – 6) / (2x^2 – 1).
The degree of the top and bottom are the same, so the limit is the coefficients, which are 4/2 (which simplifies to 2.)
Therefore, y = 2 is a horizontal asymptote.
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