I'm having trouble simplifying the result of complicated limits which contain mutliple fractions. I understand this is basic math level, but i find it difficult to find exercices to practice that specific problem.
So assuming my start limit goes like this
$$\lim_{x\to1} \frac 1 {ln(x)} - \frac {1} {(x-1)^2}$$
Form : $\infty$ - $\infty$
I put both on the same denominator
$$\lim_{x\to1} \frac {(x-1)^2 - ln (x)} {ln(x)*(x-1)^2} $$
Form : $\frac 0 0$
Hospital rule
$$\lim_{x\to1} \frac {2(x-1)- \frac 1 x} { \frac 1 x*(x-1)^2+ln(x)*2(x-1))}$$
Now i don't know if i can just cancel both $ \frac 1 x$ since there is + and -. Same goes for the $2(x-1)$.
I know that i can transform the $ \frac 1 x * (x-1)^2$ in $\frac {(x-1)^2} {x}$ but this seems like extra steps.
Thanks!