I'm questioning myselfas to why indeterminate forms arise, and why limits that apparently give us indeterminate forms can be resolved with some arithmetic tricks. Why $$\begin{equation*} \lim_{x \rightarrow +\infty} \frac{x+1}{x-1}=\frac{+\infty}{+\infty} \end{equation*} $$
and if I do a simple operation,
$$\begin{equation*} \lim_{x \rightarrow +\infty} \frac{x(1+\frac{1}{x})}{x(1-\frac{1}{x})}=\lim_{x \rightarrow +\infty}\frac{(1+\frac{1}{x})}{(1-\frac{1}{x})}=1 \end{equation*} $$
I understand the logic of the process, but I can't understand why we get different results by "not" changing anything.