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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\(^n\)
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.
(and of course is base is bigger than e, this exponential is certainly exceeds hyperbolic)
oh yes? nice
define hyperbolic function
well, it consists of some e^x componenets
cosh(x) for example is the average between e^X AND E^-X
So... `hyperbolic function` ≥ `(a)^x; where a≤e` `hyperbolic function` < `(a)^x; where a>e`
if the difference is the base, then yes
yes, with same x exponent... tnx (I was just thinking of different growths)
you can wolfram alpha it to make sure
test it with one concrete condition where a>e
I started this proposal when I proved that e^x is greater than cosh(x)
saying, as x-> infinity
then sinh(x) is certainly smaller
-e^x is always >0 e^x is always <0 :)
-e^-x is always >0, (correction). but that is always >0 as well.
wwhat wouldn't have?
other functions are also obviously less than e^x.
So I want to change my statment a bit
`hyperbolic function` < `(a)^x; where a≥e ` `hyperbolic function` > `(a)^x; where a