Editing Limit of a function (section)

===Examples===

====Non-existence of one-sided limit(s)====
[[Image:Discontinuity essential.svg|thumb|right|Function without a limit at an [[Classification of discontinuities|essential discontinuity]] ]]
The function  
<math display=block>f(x)=\begin{cases}
\sin\frac{5}{x-1} & \text{ for } x<1 \\
0 & \text{ for } x=1 \\[2pt]
\frac{1}{10x-10}& \text{ for } x>1
\end{cases}</math>
has no limit at {{math|1=''x''{{sub|0}} = 1}} (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function, see picture), but has a limit at every other {{mvar|x}}-coordinate.

The function 
<math display=block>f(x)=\begin{cases}
1 &  x \text{ rational } \\
0 &  x \text{ irrational }
\end{cases}</math>
(a.k.a., the [[Dirichlet function]]) has no limit at any {{mvar|x}}-coordinate.

====Non-equality of one-sided limits====
The function 
<math display=block>f(x)=\begin{cases}
1 & \text{ for } x < 0 \\
2 &  \text{ for } x \ge 0
\end{cases}</math>
has a limit at every non-zero {{mvar|x}}-coordinate (the limit equals 1 for negative {{mvar|x}} and equals 2 for positive {{mvar|x}}). The limit at {{math|1=''x'' = 0}} does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).

====Limits at only one point====
The functions
<math display=block>f(x)=\begin{cases}
x &  x \text{ rational } \\
0 &  x \text{ irrational }
\end{cases}</math>
and
<math display=block>f(x)=\begin{cases}
|x| &  x \text{ rational } \\
0 &  x \text{ irrational }
\end{cases}</math>
both have a limit at {{math|1=''x'' = 0}} and it equals 0.

====Limits at countably many points====
The function 
<math display=block>f(x)=\begin{cases}
\sin x &  x \text{ irrational } \\
 1     &  x \text{ rational }
\end{cases}</math>
has a limit at any {{mvar|x}}-coordinate of the form <math>\tfrac{\pi}{2} + 2n\pi,</math> where {{mvar|n}} is any integer.