infinity divided by itself sometimes? =0

[breezer]

I seem to remeber coming across this somewhere in college (10 years ago) in Calculus. I am a high school math teacher and would like to know if infinity divided by itself equals zero. If so, when and why? Under what circumstances? I do know that it is supposed to equal one.

[Tormod]

Not being a math wizard, I would assume that we would need to know what kind of infinity we are talking about.

[tom]

Infinity isn't a number so you can't use math operators on it. If you're talking about a limit , that's something different. The limit could be 0, 1 , or any other number ...
infinity / infinity is a case of indetermination.

lim x ^ (n-1) / x ^ n then x goes to infinity

is equal to infinity / infinity. When you simplify with x ^ (n-1)

lim x ^ (n-1) / x ^ n = lim 1 / x which is 0

lim x ^ n / x ^ n = lim 1 =1

lim 2 * x / x = 2

lim x ^ 2 / x = infinity

I know this aren't the best examples , but I hope it's helpful.

[maddog]

Originally Posted by breezer

I seem to remeber coming across this somewhere in college (10 years ago) in Calculus. I am a high school math teacher and would like to know if infinity divided by itself equals zero. If so, when and why? Under what circumstances? I do know that it is supposed to equal one.

Tom is basically right. The representation of infinity divided by infinity is not a number per se. However,
you can take the ratio of two divergent series and if the denominator diverses at a less rate than the
numerator you could have a converging ratio in the limit.

Tom's examples were great.

To test for convergence you use L'Hospital's Rule (I learned in second semester Calculus) by taking the
derivative separately of both numerator and denominator. If the derivatives converge then the ratio
will converge. Neat huh...

Maddog

[tom]

Originally Posted by maddog

To test for convergence you use L'Hospital's Rule

or you could show it's Cauchy sequence , or use this ( I don't know it's name in English)

If |Xn - X |<= an for every n and an -> 0 then xn -> x

[Tim_Lou]

the graph of x/x is simply a line of y=1, so, graphically, it approaches to 1 as x approaches infinity.

well, tom is right, it depends on the increasing rate of the denominator and numerator....
if the information given is simply infinity over infinity, i would say it is undefined.

[Tim_Lou]

oh wait, you say "infinity divided by itself"
which means it is x/x
hehe, using L'Hospital's Rule:
lim x--> infinity (x/x) = lim x-->infinity (1/1) = 1

[tom]

Infinity divided my itself doesn't mean x/x
is could mean 2 * x / x or x+1 /x or x ^ 2 / x

[Tim_Lou]

by itself, it implies that the "infinity" and the "thing" that divides it are identical, therefore it is x/x

lim x--> infinity 2x / x can be infinity divided by infinity, but it is definitely not something divided by itself, since 2x and x approaches infinity differently.

[little cloud]

Isn't anything divided by itself equal to 1? Would that mean that two divided by two is the same as infinity divided by itself?