

The
limit of a function 
The
definition of the limit of a function 
A limit
on the left (a lefthand limit) and a limit
on the right (a righthand limit) 
Continuous
function 
Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) 
Infinite limits 
The
limit of a function examples 
Vertical, horizontal
and slant (or oblique) asymptotes 





The
definition of the limit of a function 
The
limit of a function is a real number L
that f
(x)
approaches as x
approaches a given real number a,
written 

if
for any e
> 0 there is a d(e)
> 0 such that

f
(x)

L  < e
whenever  x

a  < d(e). 
The
definition says, no matter how small a positive number e
we take, we can find a positive number d
such that, for an arbitrary chosen value of x
from
the interval a

d
< x
< a +
d,
the corresponding function's values lie inside
the interval 
L

e
< f
(x)
< L +
e, 
as shows the right figure. 
That
is, the function's values can be made arbitrarily
close to the number L
by choosing x
sufficiently
close to
a, but not equal to
a. 



Therefore,
the number d,
that measures the distance between a point x
from the point a
on the xaxis,
depends
on the number e
that measures the distance between the point f
(x)
from the point L
on the yaxis. 
Example: Given 


whenever 


A
limit is used to examine the behavior of a function near a point
but not at the point. The function need not even be defined at
the point. 

A
limit on the left (a lefthand limit) and a limit
on the right (a righthand limit) 
The
limit of a function where the variable x
approaches the point a
from the left or, where x
is restricted to values
less than a,
is written 

The
limit of a function where the variable x
approaches the point a
from the right or, where x
is restricted to values
grater than a,
is written 

If
a function has both a lefthanded limit and a righthanded limit
and they are equal, then it has a limit at the point. Thus,
if 



Continuous
function 
A
real function
y = f (x)
is continuous at a point a
if it is defined at x =
a
and 

that
is, if for every e
> 0 there is a d(e)
> 0 such that 
f
(x)

f
(a)
 < e
whenever  x

a  < d (e). 
Therefore,
if a function changes gradually as independent variable changes,
so that at every value a,
of the independent
variable, the difference between f
(x)
and f
(a)
approaches zero as x
approaches a. 
A
function is said to be continuous if it is continuous at all
points. 

Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) 
We
say that the limit of f(x)
as x
approaches positive infinity is L
and write, 

if for any e
> 0 there exists N
> 0 such that

f
(x)

L  < e
for all x
> N
(e). 
We
say that the limit of f
(x)
as x
approaches negative infinity is L
and write, 

if for any e
> 0 there exists N
> 0 such that

f
(x)

L  < e
for all x
< N
(e). 

Not
all functions have real limits as x
tends to plus or minus infinity. 
Thus
for example, if f
(x)
tends to infinity as x
tends to infinity we write 

if
for every number N
> 0 there is a number M
> 0 such that f
(x)
> N
whenever x
> M(N). 

Infinite limits 
We
write 

if
f (x) can
be made arbitrarily large by choosing x
sufficiently close
but not equal to a. 
We
write 

if
f(x) can
be made arbitrarily large negative by choosing x
sufficiently close
but not equal to a. 

The
limit of a function examples 
Example: Evaluate
the following limits; 


Solution:
a) As x
tends to minus infinity f(x)
gets closer and closer to 0. 
As x
tends to plus infinity f(x)
gets closer and
closer to 0. Therefore, 

b) As x
tends to 0 from the left f(x)
gets larger in negative sense. 



As
x
tends to 0 from the right f(x)
gets larger in positive sense. Therefore, 


Vertical, horizontal
and slant (or oblique) asymptotes 
If
a point (x, y) moves
along a curve f
(x)
and then at least one of its coordinates tends to infinity,
while the distance between the point and a line tends to zero
then, the line is called the asymptote
of the curve. 

Vertical
asymptote 
If
there exists a number a
such that 

then
the line x
= a is the vertical
asymptote. 

Horizontal
asymptote 
If
there exists a number c
such that 

then
the line y
= c is the horizontal
asymptote. 









Calculus
contents B 



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