Sometimes in number theory and asymptotic analysis, we have functions whose expressions contain a Bachmann–Landau term such as
- Prime number theorem [wikipedia] $\pi(x) = \mathrm{Li}(x)+ O(xe^{-a\sqrt{\log x}})$, for $a>0$
- Let $\Phi(x) \overset{\text{def}}{=} \underset{n\leq x}{\sum} \phi(n)$, where $\phi$ is the Euler's totient function, we have $\Phi(x) = \frac{3}{\pi^2}x^2+ O(x\log x)$, from this book
Given such functions, how can one compute/interpret a limit at $x\to x_0$?
More generally, if we have $f(x) = g(x)+ O(h(x))$, assuming $\lim_{x\to x_0}h(x)$ exist, do we have $$\lim_{x\to x_0} f(x) = \begin{cases}O(1)\quad \text{ if } |\lim_{x\to x_0} g(x)| < \infty \text{ and }|\lim_{x\to x_0} h(x)| < \infty\\ \infty \quad \text{otherwise}\end{cases}$$