# About the $\lim_{x \rightarrow \infty}(\ln{x}-x)$

Find the $\lim_{x \rightarrow \infty}(\ln{x}-x)$.

We know that $\ln{x}=o(x)$ as ${x \rightarrow \infty}$ therefore we can guess that the limit will be $-\infty$.

Intuitively $x$ goes to infinity way faster than $\ln{x}$.

Here it is my formal proof of this:

We have that $\lim_{x \rightarrow \infty} \frac{x}{2\ln{x}}=+ \infty$ thus form the definition, $\exists a>0$ such that $x> 2\ln{x}$ forall $x>a$.

Now from this,$\forall x>a$, we deduce that $x- \ln{x} > 2 \ln{x}-\ln{x}=\ln{x} \Rightarrow \ln{x}-x < - \ln{x}$

Finally we have $\limsup_{x \rightarrow \infty} (\ln{x}-x) \leqslant - \infty$

Thus $\limsup_{x \rightarrow \infty} (\ln{x}-x)=\liminf_{x \rightarrow \infty} (\ln{x}-x)= -\infty$ .

Is my argument correct?

• Yeah, it looks good. You use improper limits (or limits on the extended real line). If this is a homework question for a rigorous analysis course, I would make sure my course covers reasoning with these, or else I’d argue without using these. But then again – if the course didn’t cover these, the question would be ill-posed anyway … Jul 29, 2017 at 12:13
• @k.stm..No this is not a homework..I just was thinking how to tackle this limit..I did not remember the trick the answerer used thus i took liminf and limsup to solve this.It is just a general question. Jul 29, 2017 at 12:18
• Yes.It's correctly done.. If $f(x)\to \infty$ as $x\to \infty$ and $g(x)=o(f(x)$ as $x\to \infty$ then $g(x)-f(x)\to -\infty$ as $x\to \infty$ because $|g(x)|\leq |f(x)|/2$ for all sufficiently large $x$. Jul 29, 2017 at 20:13

for $x>0$,

$$\ln (x)-x=x (\frac {\ln (x)}{x}-1)$$

and $$\lim_{x\to+\infty}\frac {\ln (x)}{x}=0$$

thus

$$\lim_{+\infty}(\ln (x)-x)=-\infty$$

• Yes, this is quicker; but the questioner asked whether his/her proof was correct. Jul 29, 2017 at 12:08
• @Salahamam_Fatima..That is a nice answer and a nice trick which deserves to be upvoted,but can you see if my argument is correct? Jul 29, 2017 at 12:10
• @MariosGretsas For me it seems fine. $ln x-x <-\ln x \implies lim (ln x - x)=-infty$ Jul 29, 2017 at 12:16