Limit of a sum of functions with conditions on the limits of each

Questions
1

Hello

My attention was drawn to a twitter post analysis that I do not know how to convey in Mathematica code: limits for f[x] and `fā€™[x] are to be found and not explicitly defined..

My approach so far:

Assuming[a \[Element] Reals && 
Limit[#1 + #2 & @@ {f[x], f'[x]}, x -> \[Infinity]] == a , {Limit[
f[x], ?exp], Limit[f'[x],?exp]}

leads me to nowhere. What is the code to be inserted instead of ?exp.

I doubt very much that Mathematica can verify this remarkable proof by Hardy, but certainly find a set of functions for which this proof holds. How can it be done? Thanks.
[1]: Analysis Fact on X: "If f(x) + f'(x) -> a as x -> āˆž, then f(x) -> a and f'(x) -> 0. G. H. Hardy, 1908" / X

2

What about this?

fF = (f /. Flatten[DSolve[f'[x] == a - f[x], f, x]])[x]
fFp = fF + D[fF, x]
Limit[fF, x -> Infinity]
Limit[fFp, x -> Infinity]
3

Or - basically the same -

g = a (1 - Exp[x0 - x]) + f0 Exp[x0 - x]
Limit[g, x -> Infinity]
Limit[D[g, x], x -> Infinity]
Limit[g + D[g, x], x -> Infinity]
4

Here is a counterexample:

h[x_] = Exp[-x + Floor[x]];
Plot[h[x], {x, 0, 10}]
Plot[h[x] + h'[x], {x, 0, 10}]

It is essential to assume that the function is defined and differentiable on a whole half-line, in which case you can define g[x_] := x*f[Log[x]] and argue that g'[x]->c implies g[x]/x -> c too.

5

OK. I think EVERY (well behaved) function approaching a as limit for x towards inifinity fulfills what is asked for, because if f[ x ] ā†’ a , then f[ x + h ] - f[ x ] ā†’ a - a = 0 , therefore fā€™ā€˜[ x ] ā†’ 0 and f[ x ] + fā€™[ x ] ->a.
f may even oscillate.

Some examples :

h = a + Sin[x]/x
Limit[h, x -> Infinity, Assumptions -> And[n > 0, x \[Element] Reals]]
Limit[D[h, x], x -> Infinity, 
 Assumptions -> And[n > 0, x \[Element] Reals]]
Limit[h + D[h, x], x -> Infinity, 
 Assumptions -> And[n > 0, x \[Element] Reals]]

h = (a x^n + x^(n - 2))/(x^n + x^(n - 3))
Limit[h, x -> Infinity, Assumptions -> And[n > 0, x \[Element] Reals]]
Limit[D[h, x], x -> Infinity, 
 Assumptions -> And[n > 0, x \[Element] Reals]]
Limit[h + D[h, x], x -> Infinity, 
 Assumptions -> And[n > 0, x \[Element] Reals]]

h = a + 1/(1 + x)^n
Limit[h, x -> Infinity, Assumptions -> And[n > 0, x \[Element] Reals]]
Limit[D[h, x], x -> Infinity, 
 Assumptions -> And[n > 0, x \[Element] Reals]]
Limit[h + D[h, x], x -> Infinity, 
 Assumptions -> And[n > 0, x \[Element] Reals]]
6

Yes the 3 examples you gave answer my question. However I wonder if there is a catch somewhere, Your proof is end high school math (in my country at least) and I doubt that a mathematician of the calibre of G.H Hardy would have published such a proof.. I will have to find in which article or book he published.it.
Thanks