Solving limit equations Notice the following:
all first order differential equations take on the form:
$$G(x,f(x),\frac{d}{dx}[f(x)]) = 0  $$
notice that we can replace $\frac{d}{dx}[f(x)]$ with the expression
$$ \lim_{\delta \rightarrow 0 }{\frac{f(x + \delta) - f(x)}{\delta}}$$
Which begs me to ask:
Is there a theory to handle generic equations involving limits: not just differential equations? Consider other types of limits that could exist (ex:
$$ \lim_{\delta \rightarrow 1 } \log_\delta({\frac{f(x \delta)}{f(x)}})$$
amongst others!
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Clarifying some terms:
generic = any arbitrary arithmetic expression
example:
$$2^{x + (\frac{dy}{dx})^y} = 1 + y + y^2$$
any combination of "+", "*", "^" and inverses of functions composed using those can be presented.
handle = a framework for defining the solution of arbitrary differential equations based on a class of recursively defined "primitive" equations
example, by defining the number "e" all linear differential equations can be solved,
by defining other such similar constants all quadratic differential equations can be solved...
thus: is there a theory of: what mathematical expressions need to be defined to solve a given equation and then what is the solution in terms of these derived mathematical expressions.
 A: I'm not sure what "handle" and "generic" mean, so I cannot give a complete answer.  But your example can be turned into the left-hand side of a differential equation.  Namely, assuming $f$ is differentiable, as $\delta \to 1$, we have:
$$ f(x\delta) = f(x + x(\delta-1)) = f(x) + f'(x)x(\delta-1) + o(\delta-1) $$
where $o$ is small $o$ notation.  Therefore your limitand is:
$\displaystyle \log_\delta \left( \frac{f(x\delta)}{f(x)}\right) = \frac{\log \left( \frac{f(x\delta)}{f(x)} \right)}{\log \delta} = \frac{\log \left( \frac{1}{f(x)}\left(f(x) + f'(x)x(\delta-1) + o(\delta-1)\right) \right)}{\log \delta} = \frac{\log\left(1 + \frac{f'(x)x}{f(x)}(\delta-1) + o(\delta-1)\right)}{\delta-1 + o(\delta-1)} = \frac{\frac{f'(x)x}{f(x)}(\delta-1) + o(\delta-1)}{\delta-1 + o(\delta-1)} = \frac{f'(x)x}{f(x)} + o(1) $
I have assumed that $f(x) \neq 0$, and I use $\log$ to denote the natural logarithm, which satisfies $\log(1+\epsilon) = \epsilon + o(\epsilon)$ as $\delta \to 1$.  Taking now the limit as $\delta \to 1$ gives $xf'(x)/f(x)$.
