Examples Limit of a function

-



1 examples

1.1 non-existence of one-sided limit(s)
1.2 non-equality of one-sided limits
1.3 limits @ 1 point
1.4 limits @ countably many points





examples
non-existence of one-sided limit(s)

the function without limit, @ essential discontinuity


the function







f
(
x
)
=


{



sin
⁡


5

x
−
1






 for 

x
<
1




0



 for 

x
=
1






0.1

x
−
1






 for 

x
>
1








{\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\text{ }}x<1\\0&{\text{ }}x=1\\{\frac {0.1}{x-1}}&{\text{ }}x>1\end{cases}}}



has no limit @




x

0


=
1


{\displaystyle x_{0}=1}

(the left-hand limit not exist due oscillatory nature of sine function, , right-hand limit not exist due asymptotic behaviour of reciprocal function), has limit @ every other x-coordinate.


the function







f
(
x
)
=


{



1


x

 rational 





0


x

 irrational 









{\displaystyle f(x)={\begin{cases}1&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}



(the dirichlet function) has no limit @ x-coordinate.


non-equality of one-sided limits

the function







f
(
x
)
=


{



1



 for 

x
<
0




2



 for 

x
≥
0








{\displaystyle f(x)={\begin{cases}1&{\text{ }}x<0\\2&{\text{ }}x\geq 0\end{cases}}}



has limit @ every non-zero x-coordinate (the limit equals 1 negative x , equals 2 positive x). limit @ x = 0 not exist (the left-hand limit equals 1, whereas right-hand limit equals 2).


limits @ 1 point

the functions







f
(
x
)
=


{



x


x

 rational 





0


x

 irrational 









{\displaystyle f(x)={\begin{cases}x&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}



and







f
(
x
)
=


{




|

x

|



x

 rational 





0


x

 irrational 









{\displaystyle f(x)={\begin{cases}|x|&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}



both have limit @ x = 0 , equals 0.


limits @ countably many points

the function







f
(
x
)
=


{



sin
⁡
x


x

 irrational 





1


x

 rational 









{\displaystyle f(x)={\begin{cases}\sin x&x{\text{ irrational }}\\1&x{\text{ rational }}\end{cases}}}



has limit @ x-coordinate of form





π
2


+
2
n
π


{\displaystyle {\frac {\pi }{2}}+2n\pi }

, n integer.







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