
ALGEBRA
 solved problems 






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). 

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). 

Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) 


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. Thus, 



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). 


Find
the vertical and the horizontal asymptote
of the function 


Solution:
Since, 

then x
= 1 is the vertical asymptote. 



And
since, 



then y
= 2 is the horizontal asymptote. 


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. 


Find
the following limits,



Solution:
The graph of the tangent
function shows, 

as
x
approaches
p/2 from the left,
the tangent
function increases
to plus infinity, while as x
approaches
p/2 
from the
right, the function decreases to minus infinity, therefore 



Evaluate
the limit 



Solution: 




Evaluate
the limit 



Solution:
The graph of the
arctangent
function shows,
as x
tends to minus infinity the function values
approach 
p/2
while, as x
tends to plus infinity,
the function values approach p/2. 
Therefore, 

and 














Solved
problems contents  A 



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