Limits at infinity Limit of a function

the limit of function @ infinity exists.

for f(x) real function, limit of f x approaches infinity l, denoted

lim

x
→
∞


f
(
x
)
=
l
,


{\displaystyle \lim _{x\to \infty }f(x)=l,}

means



ε
>
0


{\displaystyle \varepsilon >0}

, there exists c such




|

f
(
x
)
−
l

|

<
ε


{\displaystyle |f(x)-l|<\varepsilon }

whenever x > c. or, symbolically:

∀
ε
>
0

∃
c

∀
x
>
c
:


|

f
(
x
)
−
l

|

<
ε


{\displaystyle \forall \varepsilon >0\;\exists c\;\forall x>c:\;|f(x)-l|<\varepsilon }

.

similarly, limit of f x approaches negative infinity l, denoted

lim

x
→
−
∞


f
(
x
)
=
l
,


{\displaystyle \lim _{x\to -\infty }f(x)=l,}

means



ε
>
0


{\displaystyle \varepsilon >0}

there exists c such




|

f
(
x
)
−
l

|

<
ε


{\displaystyle |f(x)-l|<\varepsilon }

whenever x < c. or, symbolically:

∀
ε
>
0

∃
c

∀
x
<
c
:


|

f
(
x
)
−
l

|

<
ε


{\displaystyle \forall \varepsilon >0\;\exists c\;\forall x<c:\;|f(x)-l|<\varepsilon }

.

for example

lim

x
→
−
∞



e

x


=
0.



{\displaystyle \lim _{x\to -\infty }e^{x}=0.\,}