Limit of f(x) at x=a

 


A function   f(x)  is said to tend to the limit   l   at   x=a  

if given   element>0  there exists a number  delta>0


such that ;  /f(x)-l/< that element 


whenever ; /x-a/<delta 


In simple words ,   f(x)  is said to tends to the


limit l  at  x=a 


if   f(x)  tends to  l  as  x  approaches  a  


through values of  x    greater then   a 


as well as   values of  x  smaller than  a.


The limit l  of  f(x)  at  x=a  denoted by


lim x-->a  f(x)=l


It is clear from above that


f(x)  tends to the limit  l  at  x=a  implies .


lim x-->a+0  f(x) = lim x-->a-0  f(x) 

=lim x-->a  f(x) = l

provided both the right-hand and left-hand limits exists for

x=a  .

If the domain of definition of   f(x)  is

[a, b] , then existence of

lim x-->a+0  f(x)   does not arise .

Similarly the question of existence

 lim x-->b+0  f(x)   also does not arise .