Find an appropriate $ f( x)$ for each of the following.
\begin{alignat*}{6}
a) &\lim_{x \to 2}{ f( x)} &&= 0, & f( 2) &=\frac{0}{ 0} &
b) &\lim_{x \to -1}{ f( x)}&&= 2, & f( {-1}) &=\frac{0}{ 0} \\
c) &\lim_{x \to \pi}{ f( x)} &&= \infty, & f( {\pi}) &=
\infty \cdot 0 &
d) &\lim_{x \to 1}{ f( x)} &&= 1, & f( {1}) &=\frac{0}{ 0} \\
e) &\lim_{x \to \pi}{ f( x)} &&= 0, & f( {\pi}) &=
\infty \cdot 0 &
f) &\lim_{x \to d}{ f( x)} &&= 3, & f( {d}) &=\infty \cdot 0
\end{alignat*}
Do you notice anything strange about the limit of $ f( x)$ as $x \to c$ when
$ f( c)$ has the form $\infty \cdot 0$?
Tags: limits, limits-unbounded, abstract-function, creating-examples, a--a, a--v, generalization, t2, c2
This problem addresses a gnawing problem with students: they want to treat infinity times zero and zero over zero with some kind of algebraic rule of thumb. It gets worse if students use, say 0/0=1 and get the right answer for a problem; it becomes that much harder to dislodge them because they hold (in their minds) the trump: they got the right answer. So this kind of problem is a way to get them thinking actively about the ambiguity of the statements, which could well be more effective than showing them sqrt(x)/x from thin air, etc.
The “strangeness” referred to is probably meant to be the fact that it can equal any arbitrary value.
– Eric Hsu 1998-08-09
This is a fun problem! Students will need guidance. You may have to give a few hints, but let them think about the problem first - it is meant to cause some dissonance.
– James Epperson 1998-08-09
*I have a big problem with these. The use of expressions like f(c)=infty x 0 on a worksheet suggests to students that they can treat infinity like a real number and all the properties of real numbers hold. When my students write expressions like these I tell them that they don’t have any meaning, so I certainly wouldn’t have these “equalities” on a worksheet!*
Why is there no "Limits -indeterminate forms" keyword?
– Concha Gomez 1999-02-19