<s>[INST] Evaluate $f(x)=x\log{\left|\frac{x+2}{3-x}\right|}$ for $x\rightarrow\infty$ I have the following function:
$$f(x)=x\log{\left|\frac{x+2}{3-x}\right|}$$
I want to find the limit for $x\rightarrow+\infty$.
This is what I do. Since $x>=0$, I can remove the absolute value:
$$f(x)=x\log\left({\frac{x+2}{3-x}}\right)\sim x\left( \frac{x+2}{3-x}-1\right)=x\left(\frac{2x-1}{3-x}\right)=x\left(\frac{2x}{-x}\right)=-2x\rightarrow-\infty$$
The textbook reports that the limit is actually $5$. Why is my solution wrong?
 [/INST] We have for big $x > 0$
$$
\ln \left|\frac{x+2}{3-x}\right|^x = \ln \left(\frac{x+2}{x-3}\right)^x
$$
and also
$$
\lim_{x\to \infty}\left(\frac{x+2}{x-3}\right)^x = e^5
$$
hence
$$
\lim_{x\to\infty}x\ln\left|\frac{x+2}{3-x}\right| = 5
$$
</s>