Another example with a horizontal asymptote is
knowledge as a function of time -- #51, p. 222. We
might guess that accumulated knowledge in studying for
an exam looks something like this:
We might imagine that this physical process becomes less
productive from hour to hour as the evening wears on (the law
of diminishing returns).
- The hyperbola: $f(x)=\frac{1}{x}$
More generally, If $r>0$ is a rational number, then
$\lim_{x \to \infty} \frac{1}{x^r}=0$
If, furthermore, $x^r$ is defined for negative numbers, then
$\lim_{x \to -\infty} \frac{1}{x^r}=0$
- The most general form of an hyperbola is this: $f(x)=\frac{ax+b}{cx+d}$
- What is its horizontal asymptote?
- Where is its vertical asymptote?
- A rational function will have a horizontal asymptote
any time that the degree of the denominator equals or
exceeds that of the numerator: if
$r(x)=\frac{p(x)}{q(x)}$
then if the degree of q exceeds that of
p, there is a horizontal asymptote, and the
value of the asymptote is given by examining the
approximating function given by the ratio of leading
terms alone.
- Interestingly enough, if the degree of the numerator ($n$)
exceeds that of the denominator ($m$), then the rational function
ultimately looks like a polynomial of degree $n-m$: for
example, consider the function
$f(x)=\frac{x^2-1}{x}$. Its graph will ultimately look a lot
like the function $g(x)=x$ (which one can determine by
(gasp!) long division....
This is called a "slant asymptote" (not a horizontal asymptote,
for obvious reasons! If you get far from the origin, then the difference
between the two functions falls away.
We can then replace the more complicated with the simpler.
We use this idea in physics all the time: we assume that gravity is
constant at the surface of the Earth. In fact, it varies as distance to
the center of the Earth, but we're so far away that we can take this as
a constant (its value at about 4000 miles -- our distance to the center
of the Earth). We're far enough from the origin (the center of the
Earth) that we treat acceleration due to gravity as locally constant.
- The usual rules of limits apply, with the caveat that one
must be careful when dealing with $\infty$. It
is not a number, and cannot be added to anything else (e.g. it
is not valid to write that $\infty-\infty=0$ -- for example,
$x^2$ and $x$ both go to infinity, but their difference,
$x^2-x$, also goes to infinity).
We can say, however, that $\infty*\infty=\infty$, that
$\infty*1=\infty$, that $\infty/1=\infty$, etc. So some
of the usual rules apply (and hopefully make sense!).
