Definition of indeterminate form

An indeterminate form takes place if by knowing the limits of the functions involved we cannot determine what is the overall limit. We have to do a further analysis to solve this kind of situations.

The main indeterminate forms are: $(+\mathrm{\infty })-(+\mathrm{\infty })$, $0\cdot (\pm \mathrm{\infty })$, $\frac{0}{0}$, $(+\mathrm{\infty }{)}^0$, $1^{\pm \mathrm{\infty }}$, $0^0$, $\frac{\pm \mathrm{\infty }}{\pm \mathrm{\infty }}$ where all the values that appear are limits of functions.

Note that when we have things like: $$\text{If }f(x)\to \mathrm{\infty }\Rightarrow \{\begin{array}{r}{\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{1}^{f(x)}={\displaystyle \underset{x\to +\mathrm{\infty }}{lim}1=1}}\\ {\displaystyle \underset{x\to +\mathrm{\infty }}{lim}\frac{0}{f(x)}={\displaystyle \underset{x\to +\mathrm{\infty }}{lim}0=0}}\\ {\displaystyle \underset{x\to +\mathrm{\infty }}{lim}f(x)\cdot 0={\displaystyle \underset{x\to +\mathrm{\infty }}{lim}0=0}}\\ {\displaystyle \underset{x\to +\mathrm{\infty }}{lim}\frac{0}{\frac{1}{f(x)}}={\displaystyle \underset{x\to +\mathrm{\infty }}{lim}0=0}}\end{array}$$ do not produce any indeterminate form.