Finding limits of rational logarithmic functions

Question

How to calculate $$\lim\limits_{x\to 1}\frac{\ln (1+x(x-2))}{x(x-2)}=∞$$

Answer 1

Hint:

Observe that

$$\log(1+x(x-2))=\log(x-1)^2=2\log|x-1|$$

Answer 2

Well the numerator goes to $-\infty$ because $1+x(x-2)\to 0$ while the denominator goes to $-1$ and the fraction has $+\infty$ as à limit when $x\to 1$