

The
limit of a function 
The
definition of the limit of a function 
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. 

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. 

Slant
or oblique asymptote 
If
there exist limits 

then,
a line y = mx
+ c is the slant
asymptote of the function f
(x). 

Example: Find
the vertical and the horizontal 
asymptote
of the function 

Solution:
Since, 

then x
= 1 is the vertical asymptote. 



Since, 



then y
= 2 is the horizontal asymptote. 

Example: Calculate
asymptotes and sketch the graph of the function 


Solution:
By
equating the numerator with zero and solving for
x
we
find the xintercepts, 
x^{2}

x 
2 = (x + 1)(x 
2) = 0, 
x_{1}
= 1
and
x_{2}
= 2. 
We
calculate f
(0)
to find the yintercept, 
f
(0)
= 2/3. 
By
equating the denominator with zero and solving for
x
we find the vertical
asymptote, 
x
= 3. 
Let
calculate following limits 

to
find the slant asymptote y
= mx + c. 





Therefore,
the line y
= x + 2
is the slant asymptote of the given function. 









Calculus
contents B 



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