HORIZONTAL AND VERTICAL ASYMPTOTES OF RATIONAL FUNCTIONS

Vertical Asymptote :

The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if

How to find vertical asymptote?

The graph of y = f(x) will have vertical asymptotes at those values of x for which the denominator is equal to zero.

Horizontal Asymptote :

The horizontal line y = b is called a horizontal asymptote of the graph of y = f(x) if either The graph of y = f(x) will have at most one horizontal asymptote. It is found according to the following :

How to find vertical and horizontal asymptotes of rational function?

How to find horizontal asymptote?

1) If

degree of numerator > degree of denominator

then the graph of y = f(x) will have no horizontal asymptote.

2) If 

degree of numerator  =  degree of denominator

then the graph of y = f(x) will have a  horizontal asymptote at y = a_n/b_m.

3)  If

degree of numerator < degree of denominator

then the graph of y = f(x) will have a horizontal asymptote at y = 0 (i.e., the x-axis).

Find the vertical and horizontal asymptotes of the functions given below.

Example 1 :

f(x)  =  -4/(x² - 3x)

Solution :

Vertical asymptotes : 

x² - 3x  =  0

x(x-3)  =  0

x  =  0 and x  =  3

So, the vertical asymptotes are x  =  0 and x  =  3.

Horizontal asymptotes :

Comparing highest exponents,

denominator > numerator 

So, horizontal asymptote is at y = 0.

Example 2 :

f(x)  =  (x-4)/(-4x-16)

Solution :

Vertical asymptotes : 

-4x-16  =  0

-4(x+4)  =  0

x  =  -4

So, vertical asymptote is x  =  -4.

Horizontal asymptotes :

Comparing highest exponents,

denominator = numerator 

Horizontal asymptote 

=  Coefficient of x of numerator/Coefficient of x in the denominator

y  =  -1/4

So, horizontal asymptote is y  =  -1/4.

Example 3 :

f(x)  =  (x+4)/(-2x-6)

Solution :

Vertical asymptotes : 

-2x-6  =  0

-2x  =  6

x  =  -3

So, vertical asymptote is x  =  -3.

Horizontal asymptotes :

Comparing highest exponents,

denominator  =  numerator 

Horizontal asymptote 

=  Coefficient of x of numerator/Coefficient of x in the denominator

y  =  -1/2

So, horizontal asymptote is y  =  -1/2.

Example 4 :

f(x)  =  (x³-9x)/(3x²-6x-9)

Solution :

Vertical asymptotes : 

3x²-6x-9  =  0

3(x²-2x-3)  =  0

(x-3)(x+1)  =  0

x  =  3 and x  =  -1

So, vertical asymptotes are x  =  3 and x  =  -1.

Horizontal asymptotes :

Comparing highest exponents,

numerator > denominator

So, there is no horizontal asymptote.

Example 5 :

f(x)  =  (3x²-12x)/(x²-2x-3)

Solution :

Vertical asymptotes : 

x²-2x-3  =  0

(x-3)(x+1)  =  0

x  =  3 and x  =  -1 

So, vertical asymptotes are x  =  3 and x  =  -1.

Horizontal asymptotes :

Comparing highest exponents,

numerator  =  denominator

coefficient of x from the numerator/coefficient of x in the denominator

y  =  3/1

y  =  3

So, the horizontal asymptote is y  =  3.

Example 6 :

f(x)  =  (x³-16x)/(-4x²+4x+24)

Solution :

Vertical asymptotes : 

-4x²+4x+24  =  0

-4(x²-x-6)  =  0

(x-3)(x+2)  =  0

x  =  3 and x  =  -2

Horizontal asymptotes :

Comparing highest exponents,

numerator  >  denominator

So, there is no horizontal asymptote.

Example 7 :

f(x)  =  (x²+2x)/(-4x+8)

Solution :

Vertical asymptotes : 

-4x+8  =  0

-4(x-2)  =  0

x  =  2

So, vertical asymptote is x  =  2.

Horizontal asymptotes :

Comparing highest exponents,

numerator  >  denominator

So, there is no horizontal asymptote.

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