I had the following question in an exam. I wasn't able to solve the problem and I don't understand the solution.
I am supposed to find the limit of the following sequence:
$$ \lim_{x\to 1} \frac{\ln(x) - \sin(\pi x)}{\sqrt{x -1}} $$
where x > 1.
Now the solution proceeded as follows
$$ \ln(x) = x - 1 + O(|x-1|^2) $$
which didn't make sense. I know that the taylor expansion of the natural logarithm is $ \ln(x+1) = x + O(x^2) $. But why did they add the -1 and why replace x with (x-1)?
They then set:
$$ \sin(\pi x) = -\pi (x-1) + O(|x-1|^2) $$ which again isn't very clear to me. The taylor expansion of sin(x) is: $ \sin(x) = x + O(x^3) $. So how did they get to this expansion and how should I proceed to reach the same results?
Thank you