How to Calculate Vertical Asymptote

Table of Contents

Vertical asymptote defines the domain of a function. In essence, what a vertical asymptote does is to help a student, researcher, or predictor know the value/rate that a function becomes undefined or, in layman’s terms seize to exist. Mathematically a function becomes undefined when it is denominator becomes zero. For example, f(x) = 5/0 is undefined. 

So why is it important to find the vertical asymptote? The vertical asymptote tells the mathematician the bounds or limits of a function. If the function goes past that limit, whether negative, positive, or both, then it will be exceeding its domain. 

Summarily, a vertical Asymptote highlights the value where a function exceeds its domain and as such cannot exist. 

Functions play a very important role in real-life scenarios and can be used in different situations to establish a foundational formula that determines the solutions of problems that have similar behaviors.

The Vertical Asymptote Formula

A line x = v  is the vertical asymptote of a graphed function y = ƒ(x) if 

f(x) = \pm \infty

The two presentations explain that as x approaches a positive or negative value (v) the function f(x) becomes undefined. This simply means that the value v is outside the bounds of the function, and the function will become undefined (±∞) if x becomes a. hence, A vertical Asymptote is a value that makes a rational function undefined. 

How to Find Vertical Asymptote

  • Finding the vertical Asymptote function simply is explained in the following steps.
  • Write out the function and make the denominator of the function having x = 0
  • Solve for x in the denominator.
  • The value arrived at is the Vertical Asymptote
  • The value must be a real number. A complex number means there is no vertical asymptote, and the function is boundless.

What Is Vertical Asymptote

Since one of the easiest ways to make a mathematical function undefined is to make its denominator equal to zero, the aim is to always ensure there is a denominator with x so that the vertical asymptote can be computed.

Several examples will be considered to show how to calculate a Vertical Asymptote.

Example 1

F(x) = \frac{x+5}{x-6}

According to the vertical Asymptote Calculation, the focus is on the denominator, which must have an X and be made equal to zero.

∴ VA is X – 6 = 0 

Move 6 to the other side

X = 6

The vertical asymptote is 6 and is the value where the function f(x) is undefined.

Therefore, the f(x) value bounds are all real numbers lesser than 6

Note that this solution only worked with the denominator even though there was x is the numerator. This is because there is no way the x can simplify the denominator. When it is possible for a division to occur between the x in the numerator and the x in the denominator, then steps should be taken to simplify them.

Example 2

Find the vertical asymptote of this function below

F(x) = \frac{9}{2x-16}

VA is 2x – 16 = 0

2x = 16

X = 16/2 = 8

The vertical asymptote is 8, and the value where the function f(x) is undefined.

Therefore, the f(x) value bounds are all real numbers lesser than 6

A little more complex examples will be considered

Example 3

F(x)=\frac{10x}{x^2{-25}}

Find the vertical asymptote of the function 

VA is X2 – 25 = 0

X2 = 25 

Take the square root of both side to eliminate the square

X = ±5

∴ X = 5 and X = -5 

The Vertical Asymptote is therefore -5 and 5, and that means;

The f(x) value bounds are -5<X<5.

Example 4

F(x)=\frac{x}{2x^2{+3x}}

Calculate the Vertical Asymptote

Since there is an x at the numerator, considerations to simplify the denominator with it should always be in the back of the mind of the mathematician

F(x)=\frac{x}{2x^2{+3x}}
F(x)=\frac{x}{x(2x{+3)}}
F(x)=\frac{1}{2x{+3}}

Now find the VA

VA is 2x + 3= 0

2x = 3

X = 3/2

The vertical Asymptote is 3/2

Example 5

Find the Vertical Asymptote of the function below

F(x)=\frac{x^2{-3x}}{x^2{+4x}-21}

Simplify the numerator and denominator

The numerator will be:

x^2{-4x}=x(x-3)

The denominator is a quadratic equation and will be solved in that respect

x^2{+4x -21}=x^2{-3x}+7x-21

(x^2-3x) +(7x-21)

x(x-30)-7(x+3)

(x-3)(x+3)

The denominator has been simplified. The function as such will be

F(x)=\frac{x(x-3)}{(x-3)(x+3)}=\frac{x}{x-3}

now we calculate the VA

VA = x – 3 = 0

X = 3

The vertical Asymptote is 3/2

Example 6

Find the Vertical Asymptote of the function and determine its bounds of real numbers

F(x)=\frac{9}{x^2{+4}}

The VA will be x2 + 4 = 0

x2 = -4 

Usually, the next step would be to take the square root of both sides. However, since the -4 is not positive, it would be impossible to get a real number as the square root. The solution will therefore be

x=\sqrt{4x-1}

x=\sqrt{4}x\sqrt{-}1

x=2\sqrt{-}1

Now, the -1 is not a Real Number. However, it can be simplified according to the rule of complex numbers. In complex number, -1 = i

Now, the \sqrt{-1} is not a Real Number. However, it can be simplified according to the rule of complex numbers. In complex number, \sqrt{-1}=i

Therefore, x = 2i

VA = 2i

Since the VA is a complex number instead of a Real number, then the Vertical Asymptote for this function. This also implies that this function has no bounds. 

The Implication of Complex numbers in calculating Vertical Asymptotes?

As already stated in this article, once the value of a Vertical Asymptote is a real number, it means that there is no value to which the function is bounded. As such, that function is either boundless or inaccurate. In most cases, it would be considered boundless.

The implication of the complex number highlights the need for a function Vertical asymptote to be a Real number. 

Conclusion

Calculating the vertical asymptote of a function is not exactly difficult. However, understanding while it must be determined to give significance to its use. Functions have for long been used to determine trends and predict results on patterns that follow the same behavior. 

Nevertheless, there are certain instances where the bounds of a Rational function may not be clear and, as such, make it difficult for predictions to be possible. The Vertical Asymptote can help give clarity on these limits and determine the value or values at which the function ceases to exist. These Vertical asymptotes also determine if the function is boundless. When a Vertical asymptote includes a complex number, then none mathematically exists for that function.

In this article, the Vertical asymptote was defined and expressed with several examples. Following the steps for solving the equation makes it easy to determine Vertical asymptotes easily.

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