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.

If $f(x)=x$ and $g(x)=\frac{1}{x}+1$ then we know that $\displaystyle\lim_{x \to{+}\infty}{f(x)}=+\infty$ and $\displaystyle\lim_{x \to{+}\infty}{g(x)}=1$ but we cannot know beforehand the result of the limit $\displaystyle\lim_{x \to{+}\infty}{g(x)^{f(x)}}=1^{+\infty}$

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

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