Anyone ever come across a real non-textbook example of a graph with a hole in it?
In Precalc, you get into graphing rational expressions, some of which reduce to a non-rational. The cancelled factors in the denominator still identify discontinuity, yet can't result in vertical asymptotes, but holes.
Thanks!
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$\begingroup$Yes, this actually came up in a loan calculator I was asked to code. Given whole number $n > 0$ and real $r>0$, there is a formula for the geometric sum, $$ 1 + r + r^2 + r^3 + \cdots + r^{n} = \frac{1 - r^{n+1}}{1-r}.$$ This works fine when $r \neq 1$. However, it failed for $r=1$ (program crashes, and all that happy stuff), because there is a hole in the function $S(r) = \frac{1 - r^{n+1}}{1-r}$ at $r=1$. It is a removable discontinuity, but the computer didn't know that. It had to be hard-coded that: $$ 1+ r + r^2 + r^3 + \cdots + r^{n} = n+1, \quad \textrm{when $r = 1$}. $$
Hope this helps!
$\endgroup$ 3 $\begingroup$The sine cardinal function, $\dfrac{\sin\,x}{x}$. It turns up often enough in signal processing and a number of other applications.
$\endgroup$ 2 $\begingroup$A car goes 60 miles in 2 hours. So 60 miles/2 hours = 30 miles per hour.
But how fast is the car going at a particular instant? It goes 0 miles in 0 hours. There you have a hole!
It is for the purpose of removing that hole that limits are introduced in calculus. Then you can talk about instantaneous rates of change (such as the speed of a car at an instant), which is the topic of differential calculus.
$\endgroup$ 2 $\begingroup$I guess the derivative of the absolute value (on the reals) comes up in certain "actual" applications. It is undefined at $0$, and no way of plugging the hole makes it continuous. Which doesn't prevent one from defining arbitrarily a value of for instance $0$ at $0$, but it seems better to just leave the hole.
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