Infinite Limits and Asymptotes
Read this section to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1-8.
Definition of lim f(x) = K
The following definition states precisely what is meant by the phrase "we can guarantee that the values of are arbitrarily close to
by using sufficiently large values of
".
Definition: means for every given
, there is a number
so that if
is larger than
then
is within
units of
(equivalently;
whenever
.)
Solution: Typically we need to do two things. First we need to find a value of , usually depending on
. Then we need to show that the value of
we found satisfies the conditions of the definition.
(i) Assume that is less than
and solve for
.
If , then
and
(ii) For any , take
. (Now we can just reverse the order of the steps in part (i). ) If
and
, then
so
.
We have shown that "for every given , there is an
" that satisfies the definition.