RE: Why x*0 = x
December 13, 2015 at 5:12 pm
(This post was last modified: December 13, 2015 at 6:11 pm by Alex K.)
(December 13, 2015 at 5:07 pm)IATIA Wrote:
0.999... is valid
1 results in an undefined number.
The problem is basically apples and oranges.
No. It only gives you a valid answer if you don't just plug in .9999... but secretly pull the limit of the geometric series upfront.
If you plug in 0.999999... it is invalid because .99999...-1=0 in the denominator. (see Tibs statement below)
What you apparently would like to do, which is a different thing mathematically, is plugging in .9999999...9999 with a finite number of n digits, and only then take the limit n->infty of the whole fraction afterwards. THEN you get a finite limit for your fraction. But this is just one of many ways to calculate the limit lim_x->1 f(x) and does not mean that .999.... somehow is a different real number than 1.
I am not aware that f(.9999.....) has ever been used as a shorthand for lim_x->1 f(x), which is what you would like to define it as, as far as I can tell. It is instead f(lim_x->1 x) which is equivalent to f(1). If you want to define periodic decimals as indicating delayed limits, you are using a different terminology than any mathematics text I know. To express such things, 0+ and 0- are usually used.
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition