Limit with a big exponentiation tower

Let's observe that the limit is of form $$ f\left(x+\frac{\alpha}{f'(x)}\right) - f\left(x\right) $$ where $f(x)=e^{e^{e^x}}$ and $\alpha=e^{-a}$. Since $f$ is differentiable, we have $$ f\left(x+\frac{\alpha}{f'(x)}\right) - f\left(x\right)= \frac{\alpha}{f'(x)}f'\left(x+\xi\right) $$ for some $0\leq \xi \leq \frac{\alpha}{f'(x)}$. It remains to prove that $f$ is "continuous enough" for this limit to converge as it seems it should, that is $$ \frac{f'\left(x+\xi\right)}{f'\left(x\right)}\to 1 $$ I'll try to find some clever way to do this.

Edit: Well, brutal force will do. We have $$ \xi\leq \alpha\, e^{-e^{e^x}}e^{-e^x}e^{-x} $$ Now, $$ \frac{f'\left(x+\xi\right)}{f'\left(x\right)}= e^{e^{e^{x+\xi}}-e^{e^x}} e^{e^{x+\xi}-e^x}e^\xi $$ Obviously, $e^\xi\to 1$. Furthermore, by above inequality, $$ e^{x+\xi}-e^x=e^x\left(e^\xi - 1\right)=e^xO\left(\xi\right)\to 0 $$ and $$ e^{e^{x+\xi}}-e^{e^x}=e^{e^x}\left(e^{e^{x+\xi} -e^x}-1\right)= e^{e^x}O\left(e^{x+\xi} -e^x\right)=e^{e^x}e^x O\left(\xi\right)\to 0 $$ That finishes the proof.

Here is a very simple way using only the asymptotic expansion $e^t \in 1+t+Θ(t^2)$ as $t \to 0$

As $x \to \infty$:

  $\Large e^{e^{e^{x+e^{−\big(a+x+e^x+e^{e^x}\big)}}}}−e^{e^{e^x}}$

  $\Large = e^{e^{e^xe^y}} − e^{e^{e^x}}$ where $\large y = e^{−e^{e^x}-e^x-x-a} \in o(1)$

  $\Large \in e^{e^{e^x\big(1+y+Θ(y^2)\big)}}-e^{e^{e^x}} \subseteq e^{e^{e^x\Big(1+e^{−e^{e^x}-e^x-x-a}+Θ\big(e^{-2e^{e^x}-2e^x-2x}\big)\Big)}}-e^{e^{e^x}}$

  $\Large = e^{e^{e^x}e^z}-e^{e^{e^x}}$ where $\large z = e^{-e^{e^x}-e^x-a}+Θ\big(e^{-2e^{e^x}-2e^x-x}\big) \subseteq o(1)$

  $\Large \subseteq e^{e^{e^x}\big(1+z+Θ(z^2)\big)}-e^{e^{e^x}} \subseteq e^{e^{e^x}\Big(1+e^{−e^{e^x}-e^x-a}+Θ\big(e^{-2e^{e^x}-2e^x}\big)\Big)}-e^{e^{e^x}}$

  $\Large = e^{e^{e^x}}e^w-e^{e^{e^x}}$ where $\large w = e^{−e^{e^x}-a}+Θ\big(e^{-2e^{e^x}-e^x}\big) \subseteq o(1)$

  $\Large \subseteq e^{e^{e^x}} \large \big(1+w+Θ(w^2)\big) \Large - e^{e^{e^x}} \subseteq e^{e^{e^x}} \large \big(1+e^{−e^{e^x}-a}+Θ(e^{−2e^{e^x}})\big) \Large - e^{e^{e^x}}$

  $\Large = e^{-a} + \large Θ\big(e^{−e^{e^x}}\big)$

Notice how fast the limit converges! If higher-order error terms are desired, one just has to increase the number of terms used in the expansion of the exponential function.

Notes

This method is completely rigorous (which can be verified by explicitly finding suitable values for the constants hidden in the asymptotic notation) and in fact this is roughly how the better computer algebra systems find generalized asymptotic expansions (even if there is no Taylor series). For the same reason, this method also does not require any thinking; just work from inside out and expand whatever can be expanded, simply keeping track of the error terms using asymptotic bounds.

One natural question is how to know how many terms are needed at each expansion. One way is to simply guess and see what happens. If at some point the significant term itself is cancelled out, we can then trace backward to the places where we need an asymptotically more precise expansion. If the final result has less terms than desired, we can likewise deterministically identify the places where we need more terms in the expansion.

It's been a while since I've had to compute such limits, but the general strategy is as follows:

Let $F(x) = \exp(\exp(\exp{x})))$ and $\delta(x) = \exp(-a - x - \exp{x} - \exp(\exp{x}))$. Then we are asked to compute the limit

$$ L = \lim_{x \rightarrow \infty} F(x + \delta(x)) - F(x) $$

Notice that $\delta(x) \rightarrow 0$. By differentiability, we can write $$ F(x + \delta(x)) = F(x) + F'(x) \delta(x) + E(x)$$ where $E(x)$ is an error term (very difficult to calculate explicitly). Then $$ L = \lim_{x \rightarrow \infty} F'(x) \delta(x) + E(x).$$ Notice that $F'(x) \delta(x) = \exp(-a)$ for all $x$. Therefore it remains to prove that $$ \lim_{x \rightarrow \infty} E(x) = 0$$ By Taylor's theorem, the remainder $E(x)$ satisfies the upper bound $$ |E(x)| \le \frac{ |f''(\xi)|}{2} \delta(x)^2, $$ for some $\xi$ such that $x \le \xi \le x + \delta(x)$. After explicitly calculating $f''(x + \delta)$, then one can see that this upper bound vanishes as $x \rightarrow \infty$.