Sometimes when a function has a horizontal asymptote, we can see what it should be.
Sample Problem
Let f(x) = 4-x. Then as x approaches ∞ the function f approaches 0; there's a horizontal asymptote at y = 0. The function approaches this asymptote as x approaches ∞.
As x approaches -∞ the function f grows without bound, and therefore does not approach the asymptote, y = 0.
Sometimes it's a little challenging to see what the asymptote should be, but we're up for it. These asymptotes often appear when drawing rational functions. First we'll go through the guaranteed method that will tell us what type of asymptote we have and what it is, and then we'll show a shortcut for finding horizontal asymptotes.
The Polynomial Long Division Method
The guaranteed method is polynomial long division. This method will probably take some time, but it will get the answer.
When finding horizontal / slant / curvilinear asymptotes of a rational function, we do long division to rewrite the function. We throw away the remainder, and what is left is our asymptote. If we're left with a number, that's a horizontal asymptote (and remember, 0 is a perfectly good number). If we're left with a line of the form y = mx + b (in other words, a degree-1 polynomial), that line is our slant asymptote.
If we're left with anything else, it's a curvilinear asymptote.
The reason we throw away the remainder is that it will be a rational function whose numerator has a smaller degree than the denominator, and we know the limit of such a function as x approaches ∞ is 0.
A rational function will approach its horizontal / slant / curvilinear asymptote when x is approaching ∞ and when x is approaching -∞.
The Shortcut
Congratulations; we've survived the long division. Our reward is a shortcut for finding horizontal asymptotes of rational functions. A horizontal asymptote will occur whenever the numerator and denominator of a rational function have the same degree.
Find the horizontal asymptote of the function
.
If we do long division, we find
so the horizontal asymptote is y = 3.
Sample Problem
Find the horizontal asymptote of the function
If we do long division, we find
Therefore, the horizontal asymptote is y = 2.
Notice a pattern? We divide the leading term of the numerator by the leading term of the denominator, and that gives us the horizontal asymptote. That's it.
To summarize:
If a rational function has...
- a smaller degree polynomial in the numerator than in the denominator, that function will have a horizontal asymptote at 0. All done!
- the same degree polynomial in the numerator as in the denominator, that rational function has a horizontal asymptote which we can find by dividing leading terms only.
- a numerator one degree larger than the denominator, that rational function has a slant asymptote, which we can find by long division.
- none of the above, the function has a curvilinear asymptote, which we can find by long division.