I have to calculate some limits and try to solve them in use of taylor.
$$ \lim\limits_{x\to \infty} \left(x-x^2 \ln (1+\frac{1}{x})\right) $$
In taylor pattern I have $x_0$ to put, but there $x_0$ is $\infty$ so I want to replace it with something other $$ y = \frac{1}{x} \\ \lim_{y\to 0^+} \left(\frac{1}{y}-\frac{1}{y^2} \ln (1+y)\right) $$
Let $$ f(y) = \frac{1}{y}-\frac{1}{y^2} \ln (1+y) $$ $$f'(y) = -\frac{1}{y^2} + \left(-\frac{2}{y^3}\ln (1+y) - \frac{y^2}{1+y}\right) $$
but $f'(0)$ does not exists because I have $0$ in denominator.