\[\frac{ \infty}{ \infty}, \; 0^0, \, \infty - \infty , \; \frac{0}{0} \]
. Functions may take an indeterminate form at a point, but if a function tends to a limit at a point we may define a function to take that limiting value at the point.
\[f(x)= \frac{tan x}{x}\]
takes the indeterminate form \[\frac{0}{0}\]
at \[x=0\]
but as \[x \rightarrow 0\]
, \[tanx \rightarrow x\]
so \[lim_{x \rightarrow 0} \frac{tanx}{x}=1\]
. This limit holds as \[x \rightarrow 0\]
for \[x\]
negative and positive so the limit is well defined.Now consider the function
\[g(x)=x^x\]
. This function has indeterminate form at \[x=0\]
. As \[x \rightarrow 0\]
from the right, \[g(x)\]
tends to 1, but \[g(x)\]
is not defined for \[(-\frac{1}{2n})^{-\frac{1}{2n}}\]
.