To find oblique (or horizontal) asymptotes for rational functions:- If the degree of the numerator is equal to or larger than the degree of the denominator, then divide the numerator by the denominator (using long or synthetic division).
- At this point any fraction that remains will have a numerator whose degree is less than the degree of the denominator. As x approaches positive or negative infinity, this fraction will approach zero in value.
- If all you have is a fraction then you have a horizontal asymptote of y = 0.
- If you have more than a fraction, then your asymptote is the graph of the non-fractional part of your function.
Since the degree of the numerator is 2 and the degree of the denominator is 1 we need to divide: . I'll use dymthetic division:
-5 | 1 -8 4
---- -5 65
------------
1 -13 69
As x approaches positive or negative infinity the fraction at the end approaches zero in value. So T(x) approaches x-13 in value for these values of x. Our oblique asymptote is y = x - 13. (If there was no "x" in the non-fractional part of the divided T(x), then we'd have a horizontal asymptote.)
If you going to graph T(x), then it may be helpful to look a little more at the divided T(x). As x approaches infinity, the fraction part is a very small positive number. So T(x) is always just a little but more than x-13. So the graph of T(x) will approach the asymptote from above. And as x approaches negative infinity the fraction will be a very small negative number. So T(x) will always be a little bit less than x-13. So the graph of T(x) will approach the asymptote from below.
Here's a graph of T(x) (which shows the oblique asymptote, too):