I'm trying to figure out big $O$ notation (as $x\to\infty$ for $f(x) =\ln x$, and little $o$ notation (as $x \to 0$) for $f(x) =\ln x$.
Similarly, I am trying to find out little o notation (as $x\to 0$ and as $x\to\infty$) for $f(x) = x^2 + x$.
What I know (for $\ln x$):
- I know we have that $\ln x < x$ for all $x > 0$.
- I tried to look at the Taylor Series expansion for $x > 0$ (since I know we don't have one centered at $0$), but had no luck.
What I know (for $x^2 + x$):
- I know that $x = o(x^2)$ for all $x$
- I also know that $x^2 \in o(x^3)$
I struggle with asymptotic notation (especially little $o$); any tips to help guide me in the right direction would be much appreciated.
If $g/h\to 1$, which is a special case of $g\in O(h)$, we write $g\sim h$.
As $x\to0$, $x^2\in o(x)$ so $x^2+x\sim x$; as $x\to\infty$, $x\in o(x^2)$ so $x^2+x\sim x^2$.
As $x\to0^+$, $\ln x\to-\infty$, but it can be shown $x\ln x\to0$, so $\ln x\in o(1/x)$; as $x\to\infty$, $\ln x\in o(x^p)$ for any $p>0$. You can't get a Taylor series for $\ln x$ around $0$, but $\ln(1+x)$ is famously another story.