Asymptotic simplification

Question

If I have a function that have many $(1+o(1))$ terms, can I just simplify it by removing them and multiplying the function with one $(1+o(1))$ term? For example \begin{align} \exp(1+o(1))n^{1+o(1)}=en(1+o(1)). \end{align}

Answer 1

You cannot. If we write the $o(1)$ term as $\epsilon(n)$ where $$\epsilon(n) \xrightarrow[n \to \infty]{} 0$$ we would be claiming that

$e^{1+\epsilon(n)}n^{1 + \epsilon(n)}$ is asymptotically equivalent to $en(1+\epsilon(n))$. Note that $$\lim_{n \to +\infty} \frac{e^{1+\epsilon(n)}n^{1 + \epsilon(n)}}{en(1+\epsilon(n))} = \lim_{n \to +\infty} \frac{e^{\epsilon(n)} n^{{\epsilon(n)}}}{1+\epsilon(n)} = \lim_{n \to +\infty} n^{\epsilon(n)}$$

However, if $\epsilon(n)$ is vanishing sufficiently slowly (for instance, if, say, $\epsilon(n) = \frac{1}{\log \log n}$), then the above limit is equal to $\infty$, so in fact there is no asymptotic equivalence.