Limit of a Function at a Point
Limit of a Function at a Point
A function assigns a value given any point . We say that a function has a limit at a point if gets closer and closer to as moves closer and closer to . A more formal definition of f(x)=L is: for each , there exists a such that whenever . Note that depends on : "You give me an and I'll find you a ".
f(x)
x
f(x)
L
c
f(x)
L
x
c
lim
xc
ϵ>0
δ>0
|f(x)-L|<ϵ
|x-c|<δ
δ
ϵ
ϵ
δ
The graphic enables you to examine the limit of -1 as approaches 2. The limit is equal to 3. Note the interplay between the values and , as described above.
2
x
x
ϵ
δ