Can we apply limits piecewise in, $$\lim_{x \to 0} f(g(x)) = \lim_{x \to 0} f\left(\lim_{x \to 0} g(x)\right)$$ in the case for $$\lim_{x \to 0} \ln \left\vert(1+x)^{1/x}\right\vert = \lim_{x \to 0} \left[\ln \left\vert \lim_{x \to 0}(1+x)^{1/x} \right\vert \right] = e.$$
Seen this operation while proving the first derivative of $e^x$.
Thanks!
Too many formulas by a comment: supposing that the $n$ is a typo and you are asking about if $$\lim_{x\to 0}f(g(x)) = \lim_{x\to 0}f(\lim_{x\to 0}g(x)) $$ is true, consider: if the inner limit exists and $$\lim_{x\to 0}g(x) = a,$$ then: $$f(\lim_{x\to 0}g(x)) = f(a),$$ and in the RHS you are asking by the limit of a constant: $$ \lim_{x\to 0}f(\lim_{x\to 0}g(x)) = \lim_{x\to 0}f(a) = f(a) $$ ...
Is true that if $f$ is continuous at $\lim_{x\to 0}g(x)$: $$ \lim_{x\to 0}f(g(x)) = f(\lim_{x\to 0}g(x)) $$ (or $x\to$ whatever)