Understanding limit of exponents

[DarthMatter]

Let me try to sketch a proof in a more general case.

Theorem: Let g, h be real functions $\mathbb{R} \rightarrow \mathbb{R}$ such that $\lim_{x\rightarrow a} g(x) = L$, $\lim_{x\rightarrow L} h(x) = H$ with $L \in \mathbb{R}, H \in \mathbb{R}$. Let further $g(x) \neq L$ for all x in $\mathbb{R}$. Then $\lim_{x\rightarrow a} h(g(x)) = \lim_{x\rightarrow L} h(x)$.

Proof: For any given $\varepsilon_h > 0$, let $\delta_h>0$ be chosen such that $|h(x) - H| < \varepsilon_h$ for $0 < |x-L| < \delta_h$.
Let further $\delta_g$ be chosen such that $|g(x) - L| < \varepsilon_g=\delta_h$ for $0 <|x-a| < \delta_g$.
Then $g(x)\neq L$ implies $|g(x)-L| > 0$.
Therefore $0 < |g(x) - L| < \delta_h$, and therefore $|f(g(x))-H| < \varepsilon_h$ for $0 < |x-a| < \delta_g$ for our chosen $\delta_g$. The last inequalities show that $\lim_{x\rightarrow a} h(g(x))=H=\lim_{x\rightarrow L} h(x)$.
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Essentially, if the idea of this proof is correct, it means that $f(g(x))$ will 'spiral into' $H$ as $g(x)$ 'spirals into' L. But when $g(x)$ ever reaches $L$, you cannot be sure what happens. Probably in can also be proven in a situation where $g(x)$ may reach L, but certainly doesn't in a small intervall around a.