Asymptote (geometry)
This is an AoPSWiki Word of the Week for Nov 8-14 |
- For the vector graphics language, see Asymptote (Vector Graphics Language).
An asymptote is a line or curve that a certain function approaches.
Linear asymptotes can be of three different kinds: horizontal, vertical or slanted (oblique).
Contents
Vertical Asymptotes
The vertical asymptote can be found by finding values of that make the function undefined, generally because it results in a division by zero, which is undefined.
Be careful to distinguish this from a removable discontinuity (hole) in the graph; in a hole the factor that causes the division by zero can be canceled with another term.
Example Problem
Find the vertical asymptotes of 1) 2) .
Solution
1) To find the vertical asymptotes, must equal zero. Solving the equation:
$\begin{eqnarray*}x^2-5x&=&0\\x&=&\boxed{0,5}\end{eqnarray*}$ (Error compiling LaTeX. ! Missing \endgroup inserted.)
So the vertical asymptote is .
2) Since , we need to find where . The cosine function is zero at for all integers ; thus the functions is undefined at .
Horizontal Asymptotes
In general, to find a horizontal asymptote, take the and to find the end behavior of the function. For rational functions in the form of where are both polynomials, if the degree of the is greater than that of the degree of , then the horizontal asymptote is at . 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 . (If the degree of is less than that of , then you get a slant asymptote, explained in the next section).
Note a crucial difference between horizontal asymptotes and vertical asymptotes: a function can never be defined at a vertical asymptote, but it can be defined at a horizontal asymptote. This is because the function is undefined (division by zero) at vertical asymptotes. However, a horizontal asymptote only gives the values for the ends of the function, but doesn’t have anything to do with the behavior of the function in the “middle”.
Horizontal asymptotes also occur in the inverses of certain functions with vertical asymptotes, and can occur in rotated conics, namely hyperbolas. Then the horizontal asymptote can be found in the same method as vertical asymptotes, but in relation to instead of . For example, the hyperbola has a horizontal asymptote at .
Example Problem
Find the horizontal asymptote of .
Solution
If we take , notice that the term grows at a faster rate than the rest of the terms; hence our answer is .
Slanted Asymptotes
Slanted asymptotes are similar to horizontal asymptotes in that they describe the end-behavior of a function. For rational functions , a slanted 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 slanted asymptotes, since again the slanted asymptotes just describe the behavior of the function as approaches .
For rational functions, we can find the slant asymptote simply by long division.
Hyperbolas have two slant asymptotes. Given a hyperbola in the form of , the equation of the asymptotes of the hyperboal are at (swap if the term is positive).
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