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Can I make statements about the limit value of this function?

Anonymous Sun 28 Jun 17:51:57 2020 No.11846683 View Reply Original Report

Quoted By: >>11846689 >>11846708 >>11847487 >>11847958

Can I make statements about the limit of this function? a and b are parameters and can be all rational numbers. x is a natural number. I would appreciate help.
f(x)=((b^x)/(x^a))*(-1)^x

Anonymous

Anonymous Sun 28 Jun 2020 17:52:38 No.11846689 Report

Quoted By:

>>11846683
learn LATEX

Anonymous

Anonymous Sun 28 Jun 2020 17:55:39 No.11846700 Report

Quoted By:

limit does not exist

Anonymous

Anonymous Sun 28 Jun 2020 17:57:16 No.11846708 Report

Quoted By: >>11846823

>>11846683
limit is +-infinity if |b| is greater than 1, 0 if not

Anonymous

Anonymous Sun 28 Jun 2020 18:18:03 No.11846782 Report

Quoted By:

Can I formally prove the limit?

Anonymous

Anonymous Sun 28 Jun 2020 18:32:38 No.11846823 Report

Quoted By:

>>11846708
no
b>1 = undefined
b<1 = ?

Anonymous

Anonymous Sun 28 Jun 2020 19:57:28 No.11847142 Report

Quoted By: >>11847159

(-1)^x is not well defined for non-integer x

Anonymous

Anonymous Sun 28 Jun 2020 20:00:52 No.11847159 Report

Quoted By:

>>11847142
>x is a natural number

Anonymous

Anonymous Sun 28 Jun 2020 21:19:34 No.11847487 Report

Quoted By: >>11847613 >>11847656 >>11849736

>>11846683
Yes. The limit certainly exists whenever b^x/x^a goes to 0 (since (-1)^x is bounded). This happens when 0<b<1 (can be shown using the hospital a times plus a little extra do deal with non integer a) or when b=1 and a>0. For b>1 or when b=1 and $a\leq 0$ it doesn't exist (consider the subsequences for even and odd x, they have different limits). If b=0, it exists again and if b<0 then (helped by the fact that x is an integer) we get
$b^x(-1)^x=(-|b|)^x(-1)^x=(-1)^x|b|^x(-1)^x=|b|^x(-1)^{2x}=|b|^x((-1)^2)^x=|b|^x1^x=|b|^x$
so the limit is the same as |b|^x/x^a and we get that for 0>b>-1 it is 0, for b=-1 it is 0 if a>0, 1 if a=0, +infinity if a<0 and for b<-1 it goes to +infinity. Hope it helps anon.

Anonymous

Anonymous Sun 28 Jun 2020 21:42:35 No.11847613 Report

Quoted By:

>>11847487
Here is a more comperhensive list:
b a limit
~~~~~~~~~~~~~~~
b<-1 any +inf
-1 a<0 +inf
-1 0 1
-1 a>0 0
-1<b<1 any 0
1 a<0 doesn't exist
1 0 doesn't exist
1 a>0 0
b>1 any doesn't exist

Anonymous

Anonymous Sun 28 Jun 2020 21:50:15 No.11847647 Report

Quoted By: >>11847665

(-1)^x = e^(x*pi*i) is simply a rotation in the complex plane.

Can you make a statement about which direction it is pointing in the limit? No it keeps rotating around and around.

Can you make a statement about the amplitude? Yes.

Going from the continuous to discrete case is trivial.

Anonymous

Anonymous Sun 28 Jun 2020 21:52:25 No.11847656 Report

Quoted By:

>>11847487
If you express b in polar coordinates your argument becomes even simpler.

Anonymous

Anonymous Sun 28 Jun 2020 21:54:32 No.11847665 Report

Quoted By: >>11847668

>>11847647
x is natural number you minge

Anonymous

Anonymous Sun 28 Jun 2020 21:54:58 No.11847668 Report

Quoted By:

>>11847665
>Going from the continuous to discrete case is trivial.
Too hard for some I guess :-)

Anonymous

Anonymous Sun 28 Jun 2020 23:19:05 No.11847958 Report

Quoted By:

>>11846683
anyone who posts OP pic related is literally the worst possible human

Anonymous

Anonymous Mon 29 Jun 2020 11:24:49 No.11849736 Report

Quoted By:

>>11847487
ty anon, helps a lot.

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