I have a question about growth ranking

- idku

I have a question about growth ranking

- katieb

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- idku

So, we are taking about different growths.
From smallest to greatest
1) Polynomials
- linear growth
- quadratic polynomial (with positive coefficient)
- cubic polynomial (with positive coefficient)
- etc... (some nth degree polynomial)
2) Exponential function f(x)=a(b)\(^x\)
- this exponential eventually exceeds the polynomial
3) factorial
4) n\(^n\)

- idku

So, would I put hyperbolic functions in the exponential?
what I want to propose is that a hyperbolic function (as lim x→∞) is smaller than or equivalent to e\(^x\), and larger than any other exponential function if the base of this exponential function is less than e.

- nincompoop

yes

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- idku

(and of course is base is bigger than e, this exponential is certainly exceeds hyperbolic)

- idku

oh yes? nice

- nincompoop

define hyperbolic function

- idku

well, it consists of some e^x componenets

- idku

cosh(x) for example is the average between e^X AND E^-X

- idku

So...
`hyperbolic function` ≥ `(a)^x; where a≤e`
`hyperbolic function` < `(a)^x; where a>e`

- nincompoop

if the difference is the base, then yes

- idku

yes, with same x exponent... tnx
(I was just thinking of different growths)

- nincompoop

you can wolfram alpha it to make sure

- nincompoop

test it with one concrete condition where a>e

- idku

I started this proposal when I proved that e^x is greater than cosh(x)

- idku

|dw:1435300335244:dw|

- idku

saying, as x-> infinity

- nincompoop

correct

- idku

then sinh(x) is certainly smaller

- idku

-e^-x

- idku

-e^x is always >0
e^x is always <0
:)

- idku

-e^-x is always >0, (correction). but that is always >0 as well.

- idku

wwhat wouldn't have?

- idku

other functions are also obviously less than e^x.

- idku

So I want to change my statment a bit

- idku

`hyperbolic function` < `(a)^x; where a≥e `
`hyperbolic function` > `(a)^x; where a

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