what is something infinitly large value but not infinite multiplied by infinitly small value but not zero?

Question

when i ask my teachers what is infinity multiplied by 0 they say its indeterminant
then what would be the answer for above question

Answer 1

I don't think the expressions infinitely large value but not infinite and infinitley small value but not zero make sense. Every real number is finite and is equal to or diffrenet from zero.

The product of any real number by zero is zero. The set of the real numbers are a field with respect to addition and multiplication. To make things more convenient, especially when we deal with limits, we can extend the reals by adding the symbols oo and -oo, that is plus infinity and minus infinity. |By definition, every real number is larger than -oo and smaller than oo. Also by the usual convention, we have

oo + oo = oo
oo + or - k = oo for every real k
-oo + or - k = -oo for every real k
k * oo = oo for every positive real k
k * oo = -oo for every negative real k
0 * oo = oo * 0 = 0 (BY CONVENTION)
oo * oo = oo
and similar relations involving -oo.

I guess what you mean im your question is what happens with the product of 2 functions when one of the becomes arbitrarily large and the other arbitrarily close to zero. That is, if x-> oo , then f(x) -> oo and g(x) -> 0. The the value of the limit as x-. oo f(x) g(x) is said to be indeterminated, because, it may not exist and, if it exists, can finite or oo, depending on the functions f and g. I guess that's what your teacher meant.

Answer 2

i would think 1
1/x * x = 1

1/x approaches zero as "fast" as x approaches infinity

.1 * 10 = 1

.00001 * 100000 = 1

10^(-1000) * 10^(1000) = 1

10^(-∞) * 10^(∞) = 1

so "0" * "∞" = 1
but 0 * ∞ is indeterminate

Answer 3

When x approaches infinity, we have
(1/x)(x) = 1
(1/x^2)(x) = 0
(1/x)(x^2) = infinity
Which one do you pick?