First of all, there are no indeterminate numbers.
There are numbers and there are descriptions that sound like they might describe a number, but they do not.
"The number #x# that makes #x+3=x-5#" is such a description. As is "The number #0/0#."
It is best to avoid saying (and thinking) that "#0/0# is an indeterminate number".
.
In the context of limits:
When evaluating a limit of a function "built" by some algebraic combination of functions, we use the properties of limits.
Here are some of the. Notice the condition specified at the beginning.
If #lim_(xrarra)f(x)# exists and #lim_(xrarra)g(x)# exists,
then
#lim_(xrarra)(f(x)+g(x)) = lim_(xrarra)f(x) + lim_(xrarra)g(x)#
#lim_(xrarra)(f(x)-g(x)) = lim_(xrarra)f(x) - lim_(xrarra)g(x)#
#lim_(xrarra)(f(x)g(x)) = lim_(xrarra)f(x) lim_(xrarra)g(x)#
#lim_(xrarra)f(x) / g(x) = (lim_(xrarra)f(x)) / (lim_(xrarra)g(x))# provided that #lim_(xrarra)g(x) != 0#
Also note that we use the notation: #lim_(xrarra)f(x) = oo# to indicate that the limit DOES NOT EXIST, but we're explaining the reason (as #xrarra, #f(x) increases without bound)
If one (or both) of the limits #lim_(xrarra)f(x)# and #lim_(xrarra)g(x)# fails to exist, then the form we get from the limit properties may be indeterminate. Though it is not necessarily indeterminate.
Example 1:
#f(x)=2x+3#, and #g(x) = x^2 +x#, and #a=2#
#lim_(xrarr2)f(x) = 7# and #lim_(xrarr2)g(x) = 6# .
The value of the limit:
#lim_(xrarr2)(f(x)+g(x))# is determined by the form of the sum:
#lim_(xrarra)f(x) + lim_(xrarra)g(x) = 7 + 6#
Example 2:
#f(x)=x+3+1/x^2#, and #g(x) = x^2+7 + 1/x^2 #, and #a=0#
#lim_(xrarr0)f(x) = oo# and #lim_(xrarr0)g(x) = oo# .
In spite of the fact that neither limit exists,
the question of the limit:
#lim_(xrarr0)(f(x)+g(x))# is determined by the form of the sum:
#lim_(xrarra)f(x) + lim_(xrarra)g(x) = oo + oo = oo#
The notation looks as if we're saying somthing we're not saying. We are not saying that infinity is a number that we can add to itself to get infinity.
What we are saying is:
the limit as #x# approaches #0# of the sum of these two functions does not exist, because as #x rarr 0#, both #f(x)# and #g(x)# increase without bound, therefore the sum of these functions also increases without bound.
Example 3 : For the same set-up as example 2, consider the limit of the difference instead of the sum:
If #f(x)# and #g(x)# are increasing without bound as #x rarr 0#, we can conclude that the sum is also increasing w/o bound. But we can draw no conclusion about the difference.
#lim_(xrarr0)(f(x)-g(x))# is NOT determined by the form of the difference:
#lim_(xrarra)f(x) - lim_(xrarra)g(x) = oo - oo = "?"#
For #f-g# we eventually get # - 4#, but for #g - f# we get #+4#
Indeterminate forms of limits include:
#0/0#, #oo/oo#, #oo-oo#, #0* oo#, #0^0#, #oo ^0#, #1 ^oo#
(The last one surprised me until I got it into my memory that
#lim_(xrarroo) (1+1/x)^x = lim_(xrarr0)(1+x)^(1/x) = e#)
The form #L/0# with #L !=0# is perhaps "semi-determinate". We know that the limit fails to exist, and that it fails because of some increasing OR decreasing without bound behavior, but we can't say which.
