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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.

RobPratt
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James Smith
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  • have you tried reading on wikipedia for example ? https://en.wikipedia.org/wiki/Big_O_notation – baibo Apr 23 '20 at 20:34
  • Yes, unfortunately I have done my fair share of researching this but still cannot seem to figure it out. – James Smith Apr 23 '20 at 20:41
  • Please put dollar signs around things. – J.G. Apr 23 '20 at 21:40
  • My favorite source for these notions has been Titchmarsh, The Theory of Functions (2nd Edition), page 1. Although my use was mainly to remember which notion meant which thing, I found that his examples of each notion were quite clear. Basically, $O$ means your function has absolute value less than a constant times some specified function for a specified limit of the variable, while $o$ means the ratio of your function and a specified function has limit $0$ for a specified limit of the variable. For instance, $\ln(x)$ is $O(x)$ as $x$ becomes infinite. – Chris Leary Apr 23 '20 at 21:42
  • @ChrisLeary Thank you, I'll look into this! If ln$(x)$ is $O(x)$ as $x$ becomes infinite, would ln$(x)$ be $O(1)$ as $x$ approaches zero? – James Smith Apr 23 '20 at 22:01
  • The natural log tends to negative infinity as $x$ tends to $0,$ so I guess you could technically use any constant function, but since "a constant C" appears in the definition of $O,$ the constant function $1$ should do just fine. – Chris Leary Apr 24 '20 at 02:16

1 Answers1

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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.

J.G.
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