Limit of a Function at a Point

ϵ

1

δ

1.5

A function assigns a value

f(x)

given any point

x

. We say that a function

f(x)

has a limit

L

at a point

c

if

f(x)

gets closer and closer to

L

as

x

moves closer and closer to

c

. A more formal definition of

lim

xc

f(x)=L

is: for each

ϵ>0

, there exists a

δ>0

such that

|f(x)-L|<ϵ

whenever

|x-c|<δ

. Note that

δ

depends on

ϵ

: "You give me an

ϵ

and I'll find you a

δ

".

The graphic enables you to examine the limit of

2

x

-1

as

x

approaches 2. The limit is equal to 3. Note the interplay between the values

ϵ

and

δ

, as described above.