MHB Limits with the natural logarithm

[Yankel]

Hello

I have three limits to calculate, based on a given limits. What I know is:

\[\lim_{x\rightarrow 0}\frac{ln(1+x)}{x}=1\]

And based on this, I need to find (without L'Hopital rule), the following:

\[\lim_{x\rightarrow 0}\frac{ln(1-x)}{x}\]

\[\lim_{x\rightarrow 0}\frac{ln(1+x^{2})}{x}\]

\[\lim_{x\rightarrow 0}\frac{ln(1+2x)}{x}\]

I can't figure out the technique of moving from the known limits to the ones I need to find.

Thank you in advance !

 

[Dethrone]

Hi Yankel,

you can try using change of variables for all of them, i.e $u=-x$ for the first one, $u=x^2$ for the second one, etc.

 

[Greg]

Ok, so we're given $\lim_{x\to0}\frac{\log(1+x)}{x}=1$ and we have

$$\lim_{x\to0}\frac{\log(1-x)}{x}=\lim_{x\to0}\frac{\log(1+(-x))}{x}=-\lim_{x\to0}\frac{\log(1+(-x))}{-x}$$

Can you compute it now? Can you make progress on the others, both with this method and by using substitution as Rido12 suggested?

 

[Yankel]

So simple once you see an example...

Thank you !