Question about known limit if x is negative?

Question

We know that

$$ \lim\limits_{x \to 0} \frac{\ln(1+x)}{x}=1$$

so

$$ \lim\limits_{x \to 0} \ln(1+x)=x$$

Can I conclude that

$$ \lim\limits_{x \to 0} \frac{\ln(1-x)}{x}=1$$ and

$$ \lim\limits_{x \to 0} \ln(1-x)=-x$$

Is it correct?

Thanks

Answer 1

What you can say is that $\ln(1 + x) \sim x$ as $x\to 0$, so $\ln(1 -x) \sim -x$ as $x\to 0$. This is the proper notation.