Expert: Ahmed Salami - 10/11/2011

Question
lim x->0(x/x)=1. What will happen when I do like this...lim x->0(x)lim x->0(1/x)=?( Is the answer 1 or any other) Every limit must follow its rule... Right? Please tell me what the stuff is going on....

Answer
Hi Shameem,
You need to understand that 0 and ∞ in the study of limits have indeterminate values. They represent infinitely small and large numbers, the product of which would also be indeterminate meaning 'we cant tell exactly what it is'.
Applying the rules of limit as you tried;
lim x→0 (x) . lim x→0 (1/x) = 0 . (1/0) = 0 . ∞   which is indeterminate.

Lets see an example in numbers.
10^(-1000000) is so small that we can consider it to be 0 for some practical reasons.
10^(1000000) can be considered infinite for the opposite reason. Same goes for the number 10^(10000000).
Now,
10^(-1000000) x 10^(1000000) = 1
but
10^(-1000000) x 10^(10000000) = 10

And so you can see why 0 x ∞ is indeterminate.

Combining the limit as lim x→0 (x/x) is an exact way to go about it in this situation and we can get a true value of 1.

Regards