HOW TO FIND VERTICAL ASYMPTOTE OF A FUNCTION

Vertical Asymptote :

This is a vertical line that is not part of a graph of a function but guides it for y-values 'far' up  and/or 'far' down.

The graph may cross it but eventually, for large enough or small enough values of y, that is

y ----> ±∞

Always, the graph would get closer and closer to the horizontal asymptote without touching it.

In the diagram above, x = k is an horizontal asymptote. Because, the graph is getting closer and closer to x = k without touching it as y ----> ±∞.

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)

Rational Function - Example :

In each case, find the equation of vertical asymptote :

Example 1 :

f(x) = 1/(x + 6)

Solution :

Step 1 :

In the given rational function, the denominator is

x + 6

Step 2 :

Equate the denominator to zero and solve for x.

x + 6 = 0

x = - 6

Step 3 :

The equation of the vertical asymptote is

x = - 6

Example 2 :

f(x) = (x² + 2x - 3)/(x² - 5x + 6)

Solution :

Step 1 :

In the given rational function, the denominator is

x² - 5x + 6

Step 2 :

Equate the denominator to zero and solve for x. 

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 :

f(x) = (2x - 3)/(x² - 4)

Solution :

Step 1 :

In the given rational function, the denominator is

x² - 4

Step 2 :

Equate the denominator to zero and solve for x.

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 :

f(x) = (2x - 3)/(x² + 4)

Solution :

Step 1 :

In the given rational function, the denominator is

x² + 4

Step 2 :

Equate the denominator to zero and solve for x.

x² + 4 = 0

x² = -4

x = ±√-4

x = ±2i

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

Step 3 :

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

So, there is no vertical asymptote.

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