Yahir Tucker

2022-06-26

How to prove that $\mathrm{ln}\left(x\right) for $x\to \mathrm{\infty }$
During my calculus homework I need to prove some limits without using L'Hôpital's rule. I have difficulties to show rigorously that $\mathrm{ln}\left(x\right) for big enough x.
For example, I need to find the image of the continues function $f:R\to R$ such that for every $x\in R$: $|f\left(x\right)-x{e}^{\sqrt{|x|}}|<{x}^{4}$
I've tried to prove that $x{e}^{\sqrt{|x|}}-{x}^{4}\to \mathrm{\infty }$ if $x\to \mathrm{\infty }$, and then the image of $f$ will be all the reals. Unfortunately, I don't find the way to make it formal enough.

popman14ee

Expert

For the first part:
Note the following:
1.$\mathrm{log}\left(1\right)=0<1$
2.${\mathrm{log}}^{\prime }\left(x\right)=\frac{1}{x}\le 1=\frac{d}{dx}x$ for $x\ge 1$
3.${\mathrm{log}}^{\prime }\left(x\right)=\frac{1}{x}\ge 1=\frac{d}{dx}x$ for $x\le 1$
So you can integrate over an appropriate interval to get $\mathrm{log}\left(x\right) for all $x$.
For the second part:
Try to use
$\underset{x\to \mathrm{\infty }}{lim}\frac{{e}^{x}}{{x}^{n}}=\mathrm{\infty }$
for any natural number $n$.

preityk7t

Expert

Hint: Consider the function $f\left(x\right)=\mathrm{ln}\left(x\right)-x$. Note that $f\left(1\right)<0$ and show this function is monotonically decreasing (from $x=1$) by taking the derivative