HOW TO FIND VERTICAL ASYMPTOTE OF A FUNCTION

About "How to Find Vertical Asymptote of a Function"

How to Find Vertical Asymptote of a Function :

In this section, we are going to learn, how to find vertical asymptotes of a function. .  

And we will be able to find vertical asymptotes of a function, only if it is a rational function.

That is, the function has to be in the form of

f(x)  =  g(x) / h(x)

Example : Rational Function

Steps to Find Vertical Asymptotes of a Rational Function

Step 1 :

Let f(x) be the given rational function. Make the denominator equal to zero.

Step 2 :

When we make the denominator equal to zero, suppose we get  x  =  a and  x  =  b. 

Step 3 :

The equations of the vertical asymptotes are 

x  =  a and x  =  b

How to Find Vertical Asymptote of a Function - Examples

Example 1 : 

Find the equation of vertical asymptote for the function given below. 

f(x)  =  1 / (x + 6)

Solution :

Step 1 :

In the given rational function, the denominator is

x + 6

Step 2 : 

Now, we have to make the denominator equal to zero.

That is,                                   

x + 6  =  0

x  =  - 6

Step 3 :

The equation of the vertical asymptote is 

x  =  - 6

Example 2 :

Find the equation of vertical asymptote for the function given below. 

f(x)  =  (x² + 2x - 3) / (x² - 5x + 6)

Solution :

Step 1 :

In the given rational function, the denominator is

x² - 5x + 6

Step 2 :

Now, we have to make the denominator equal to zero.

That is,                                   

x² - 5x + 6  =  0

(x - 2)(x - 3)  =  0

x - 2  =  0 or x - 3  =  0

x  =  2 or x  =  3

Step 3 :

The equations of two vertical asymptotes are

x  =  2 and x  =  3

Example 3 :

Find the equation of vertical asymptote for the function given below. 

f(x)  =  (2x - 3) / (x² - 4)

Solution :

Step 1 :

In the given rational function, the denominator is

x² - 4

Step 2 :

Now, we have to make the denominator equal to zero.

That is,                             

x² - 4  =  0

x² - 2²  =  0

(x + 2)(x - 2)  =  0

x  =  - 2  or  x  =  2

Step 3 :

The equations of two vertical asymptotes are

x  =  - 2  and  x  =  2

Example 4 :

Find the equation of vertical asymptote for the function given below. 

f(x)  =  (2x - 3) / (x² + 4)

Solution :

Step 1 :

In the given rational function, the denominator is

x² + 4

Step 2 :

Now, we have to make the denominator equal to zero.

That is,                             

x² + 4  =  0

x²  =  - 4

x  = ± √-4

x  = ± 2i 

x  =  2i  or  x  =  - 2i   (Imaginary)

Step 3 :

When we make the denominator equal to zero, we don't get real values for "x".

Hence, there is no vertical asymptote.

After having gone through the stuff given above, we hope that the students would have understood, "How to Find Vertical Asymptote of a Function". 

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