- For the vector graphics language, see Asymptote (Vector Graphics Language).
Linear asymptotes can be of three different kinds: horizontal, vertical or slant (oblique).
The vertical asymptote can be found by finding values of that make the function undefined. Generally, it is found by setting the denominator of a rational function to zero.
If the numerator and denominator of a rational function share a factor, this factor is not a vertical asymptote. Instead, it appears as a hole in the graph.
A rational function may have more than one vertical asymptote.
Find the vertical asymptotes of 1) 2) .
1) To find the vertical asymptotes, let . Solving the equation:
So the vertical asymptotes are .
2) Since , we need to find where . The cosine function is zero at for all integers ; thus the functions is undefined at .
For rational functions in the form of where are both polynomials:
1. If the degree of is greater than that of the degree of , then the horizontal asymptote is at . This can be seen by noting that as increases, increases much faster than does. Since the denominator increases faster than the numerator, as x approaches infinity, y gets smaller until it approaches zero. A similar trend can be seen as x decreases.
2. If the degree of is equal to that of the degree of , then the horizontal asymptote is at the quotient of the leading coefficient of over the leading coefficient of .
3. If the degree of is less than the degree of , see below (slanted asymptotes)
A function may not have more than one horizontal asymptote. Functions with a "middle section" may cross the horizontal asymptote at one point. To find this point, set y=horizontal asymptote and solve.
Find the horizontal asymptote of .
The numerator has the same degree as the denominator, so the horizontal asymptote is the quotient of the leading coefficients:
For rational functions , a slant asymptote occurs when the degree of is one greater than the degree of . If the degree of is two or more greater than the degree of , then we get a curved asymptote. Again, like horizontal asymptotes, it is possible to get crossing points of slant asymptotes.
For rational functions, we can find the slant asymptote simply by long division, omitting the remainder and setting y=quotient.
Find the slant asymptote of
The slant asymptote is
3 minute asymptote lesson: