Indeterminate Forms

An indeterminate form is any expression whose value is not easily or well defined. Typical indeterminate forms are  
\[\frac{ \infty}{ \infty}, \; 0^0, \, \infty - \infty , \; \frac{0}{0} \]
.
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  
\[(-\frac{1}{2n})^{-\frac{1}{2n}}\]
.

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