Maple says that this limit is zero but I can't prove it. Any help or tips would be appreciated.


  • 2
    $\begingroup$ Hint: take logs, rearrange, L'H. $\endgroup$
    – vadim123
    Jan 28, 2015 at 19:38
  • $\begingroup$ See this. $\endgroup$ Jan 28, 2015 at 19:41

2 Answers 2


We have

$$n\left(1-\frac{1}{\ln(n)}\right)^n=n\exp\left(n\ln\left(1-\frac{1}{\ln(n)}\right)\right)\sim_\infty n\exp\left(-\frac{n}{\ln n}\right)\xrightarrow{n\to\infty}0$$ and the last limit can be proved using the L'Hôpital's rule.

  • $\begingroup$ Thanks. I see that $\displaystyle\lim_{n\rightarrow\infty}-\ln\left(1-\frac{1}{\ln(n)}\right)\ln(n)=1$ by L'Hospital's rule so this does indeed work. $\endgroup$
    – Lucas
    Jan 28, 2015 at 19:55

$$\lim_{n\to\infty}n\bigg(1-\frac{1}{\ln(n)}\bigg)^n=\lim_{n\to\infty}n\bigg[\bigg(1-\frac{1}{\ln(n)}\bigg)^{\ln(n)}\bigg]^{\frac{n}{\ln(n)}}=\lim_{n\to\infty}\frac{n}{e^{\frac{n}{\ln(n)}}}=0$$ Where I used the fact that $$\lim_{a_n\to\infty}\left(1-\frac{1}{a_n}\right)^{a_n}=e^{-1}$$ and for $\ n\to\infty$ $$\ n>>\ln(n)$$

  • $\begingroup$ happy to help, when I can $\endgroup$
    – Mosk
    Jan 28, 2015 at 21:15
  • $\begingroup$ I think it could be better to transform the expression to verify if the limit exists and if it exists then write "lim of the expression = 0" It's more speed and better I think :) $\endgroup$
    – ParaH2
    Feb 6, 2015 at 11:57
  • $\begingroup$ I don't understand your second equality, it looks as if you took the limit of the inner sequence in the brackets, and then took the limit of the entire sequence again, which is generally a false thing to do. What is your justification in this case? $\endgroup$
    – user7610
    Feb 6, 2015 at 13:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.