For the two functions $f(n)=n^{100}$ and $g(n)=2^{n/100}$, I am trying to determine whether $f(n) = O(g(n))$. In order to do this, I used L'Hopital's rule as if $\lim\limits_{n\to\infty} \frac{f(n)}{g(n)}=0$, this implies that eventually $f(n) < g(n)$, and thus $f(n) = O(g(n))$.

As the following limit is of indeterminate form, L'Hopital's rule can be used: $$\lim_{n\to\infty} \frac{f(n)}{g(n)} = \frac{\infty}{\infty}$$

However, when I tried using L'Hopital's rule using the derivatives of $f(n)$ and $g(n)$ with respect to $n$, the limit is of indeterminate form: $$\lim_{n\to\infty} {100n^{99}\over 2^{n/100-2}\cdot \ln2/25} = \frac{\infty}{\infty}$$

Is there any other method I can use to demonstrate the $f(n) = O(g(n))$, since using L'Hopital's rule did not work. Any insights are appreciated.

  • $\begingroup$ Notice that your limit is still in the indeterminate form $\infty / \infty$, and you can get your limit by repeatedly using L'Hopital. $\endgroup$ – Hyperion Mar 10 at 23:04
  • $\begingroup$ So can I use it on the expression f''(n)/g''(n), and so on, until I get a definite answer? $\endgroup$ – ceno980 Mar 10 at 23:06
  • $\begingroup$ If your limit is still in the form $\infty/\infty$ meets all the criterion for applying L'Hopital, you can use it as many times as you find necessary. So yes. $\endgroup$ – Hyperion Mar 10 at 23:07
  • $\begingroup$ @ceno980. Yes, 99 times. $\endgroup$ – William Elliot Mar 10 at 23:10
  • $\begingroup$ I don't think L'Hopital's rule will be the best way to approach this problem. $\endgroup$ – ceno980 Mar 10 at 23:11

Nor mally you shouldn't even think of L'Hospital? It is a basic result that for any $\alpha, \beta>0$, one has $$x^\alpha =o(\mathrm e^{\beta x}) \quad(x\to+\infty),\;\text{ which means }\lim_{x\to+\infty}\frac{x^\alpha}{\mathrm e^{\beta x}}=0.$$ Now, note that $2^{n/100}=\mathrm e^{\ln 2\cdot n/100}$.

  • $\begingroup$ What does o(e^βx) mean? Thanks. $\endgroup$ – ceno980 Mar 10 at 23:14
  • $\begingroup$ It means what I wrote at the end of the equation. $\endgroup$ – Bernard Mar 10 at 23:19
  • $\begingroup$ What does the o in o(e^βx) specifically mean? $\endgroup$ – ceno980 Mar 10 at 23:24
  • $\begingroup$ For me it's only a notation introduced by Edmund Landau for a class of functions at the beginning of the 20th century. It reads as ‘little oh’. $\endgroup$ – Bernard Mar 10 at 23:34
  • $\begingroup$ Would it be possible to know how the above result was derived? E.g. Is there a proof for it? $\endgroup$ – ceno980 Mar 10 at 23:42

We have $\log\left(n^{100}\right)=100\log(n)$ and $\log\left(2^{n/100}\right)=\frac{\log(2)}{100}n$. Both of these tend to $\infty$. Applying L'Hopital to the logs gives $$ \begin{align} \lim_{n\to\infty}\frac{100\log(n)}{\frac{\log(2)}{100}n} &=\lim_{n\to\infty}\frac{100\frac1n}{\frac{\log(2)}{100}}\\ &=0 \end{align} $$

  • $\begingroup$ Is the answer valid if you apply L'Hopital's rule to log(f(n)) and log(g(n))? I heard that when trying to demonstrate that f(n) is O(g(n)) you can't use the logs of f(n) and g(n) since f(n) ≠ log(f(n)). $\endgroup$ – ceno980 Mar 10 at 23:54
  • $\begingroup$ If $\lim\limits_{n\to\infty}g(n)=\infty$ and $\lim\limits_{n\to\infty}\frac{\log(f(n))}{\log(g(n))}=0$, then for $n$ large enough, $\frac{\log(f(n))}{\log(g(n))}\le\frac12$, so $\frac{f(n)}{g(n)}\le\frac1{\sqrt{g(n)}}\to0$. $\endgroup$ – robjohn Mar 11 at 1:25
  • $\begingroup$ For the OP: the key requirement here for algebraic manipulation is that the inverse of $\log$ is non-decreasing. This gives a theorem more general than for just $\log$s. $\endgroup$ – adfriedman Mar 11 at 8:00

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