<s>[INST] Limit of function with natural logarithm I need help solving this problem. I tried L'hospital and rearranging but nothing worked.
$$
\lim_\limits{x→∞} x^2\left(\ln\left(1 + \frac{1}{x}\right)- \frac{1}{x+1}\right)
$$
 [/INST] We have 
$$x(\frac{\ln (1+\frac{1}{x})}{\frac{1}{x}})-\frac{x^2}{1+x}$$
Let $x=\frac{1}{y}$. We have,
$$=\frac{1}{y}\frac{ \ln (1+y)}{y}-\frac{1}{y^2+y}$$
This is,
$$=\frac{1}{y} \frac{ \ln (1+y)}{y}-\frac{1}{y}+\frac{1}{y+1}$$
$$=\frac{1}{y}(\frac{\ln(1+y)}{y}-1)+\frac{1}{y+1}$$
$$=\frac{\ln(1+y)-y}{y^2}+\frac{1}{y+1}$$
The first limit, as $y \to 0^+$ can be computed by LH used twice.
$$ \to -\frac{1}{2}+1=\frac{1}{2}$$
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