Under which conditions can we algebraically manipulate around an $\approx$ sign?

Question

I noticed the following, and have a more general question that stems from it.

The natural logarithm can be defined as follows. $$\ln x := \lim_{\epsilon \rightarrow 0} \frac{x^\epsilon - 1}{\epsilon}$$ If we consider the case where $\epsilon \approx 0$ (for some non-rigorous definition of $\approx$), then we have $$\ln x \approx\frac{x^\epsilon - 1}{\epsilon}$$ This can then be algebraically manipulated as if $\approx$ were $=$. $$\epsilon \ln x \approx x^\epsilon - 1$$ $$1 + \epsilon \ln x \approx x^\epsilon$$ $$x \approx \left( 1 + \epsilon \ln x \right)^{1/\epsilon}$$ $$e^{\ln x} \approx (1 + \epsilon \ln x)^{1/\epsilon}$$ $$e^u \approx (1 + \epsilon u)^{1/\epsilon}$$ Turning this back into the language of limits, we have $$e^u = \lim_{\epsilon \rightarrow 0}(1 + \epsilon u)^{1/\epsilon}$$ This is, of course, true, though the math that got there is not rigorous, and wouldn't be possible if we kept the limit in place.

Here's another similar observation.

$e$ can be approximated by a Taylor series. $$e^x = \sum_{n = 0}^\infty \frac{x^n}{n!}$$ If $x \approx 0$, we can write $$e^x \approx 1 + x.$$ This can be rearranged to $$e \approx (1+x)^{1/x},$$ which in the language of limits is $$e = \lim_{x \rightarrow 0} (1+x)^{1/x},$$ which is of course a true statement. This also works if we take more Taylor terms. $$e^x \approx 1 + x + \frac{x^2}{2} \Longleftrightarrow e = \lim_{x \rightarrow 0} \left( 1 + x + \frac{x^2}{2} \right)^{1/x}$$ $$e^x \approx 1 + x + \frac{x^2}{2} + \frac{x^3}{6} \Longleftrightarrow e = \lim_{x \rightarrow 0} \left( 1 + x + \frac{x^2}{2} + \frac{x^3}{6} \right)^{1/x}$$ $$\text{et cetera}$$ As far as I can tell all of these statements are, surprisingly, true.

I have a couple questions about what's going on here:

1. What exactly is this relation $\approx$ in the way that I am using here?
2. Under what circumstances are algebraic manipulations that normally are done with $=$ instead done with $\approx$ valid?
3. Perhaps all of this is just limits in disguise. If you began with $\ln x = \lim_{\epsilon \rightarrow 0} \frac{x^\epsilon - 1}{\epsilon}$, is there a way to algebraically transform it to $e^x = \lim_{\epsilon \rightarrow 0}(1 + \epsilon x)^{1/\epsilon}$ in a mathematically sound way?

I'm basically looking for more intuition about exactly what is happening and what circumstances are allowing these hand-wavy manipulations to generate true statements from other true statements.