Difference between the value of function at A and the limit

[amaresh92]

what is the difference between the value of function at A and the limit of function at A.
to find the limit of function by direct substitution we just put the value A in function which gives the limit.but i think it should give the value of function at that point .how it become limit?

 

[radou]

A function may not be defined at some point c, but the limit, as x --> c, may exist. A function whose limit, as x --> c through the domain, exists and equals f(c) is continuous at the point c.

 

[HallsofIvy]

what is the difference between the value of function at A and the limit of function at A.
to find the limit of function by direct substitution we just put the value A in function which gives the limit.but i think it should give the value of function at that point .how it become limit?

That only works if the function is continuous. In fact it is the definition of continuous:
A function is said to be "continuous at a" if and only if $\lim_{x\to a} f(x)= f(a)$".

Otherwise, there is no relationship at all between f(a) and $\lim_{x\to a} f(x)$.


For example, "f(x)= 3x for x any number except 1 and f(1)= 5" is a perfectly valid function. It's value at x= 1 is 5 but the limit as x goes to 1 is 3.