Vertical Asymptote

A vertical asymptote is a vertical line that a function approaches but never reaches as the input value gets close to a certain number.

In other words, if the values of a function increase or decrease without bound ( +\infty or -\infty ) as x approaches a particular value ak, then the line x = k is a vertical asymptote.

A line x = k is a vertical asymptote of f(x) if at least one of the following holds:

• lim _x→k f(x) = ±∞
• lim _x→k+ f(x) = ±∞
• lim _x→k- f(x) = ±∞

Properties of Vertical Asymptote

The following are the properties of a vertical asymptote:

• The function becomes unbounded near the VA.
• The VA doesn't touch or cross the function's curve.
• The x-value of the VA is not in the function's domain.

Vertical Asymptote Rules

The rules for finding vertical asymptotes are as follows:

Rule 1: Simplify a rational function then set its denominator to zero to determine its vertical asymptotes.

Rule 2: As with linear functions, quadratic functions, cubic functions, etc., exponential and Poisson functions lack vertical asymptotes.

Rule 3: Set ax + b = 0 then solve for x to determine the vertical asymptotes of logarithmic function f(x) = log (ax + b).

Rule 4: With exception from sin x and cos x, all trigonometric functions have vertical asymptotes.

• tan x at x = πn + π/2
• cosec x at x = πn
• sec x at x = π/2 + nπ
• cot x at x = πn

Where, n is an integer.

Rule 5: To find the vertical asymptote of any other function than these, just think what values of x would make the function to be ∞ or -∞.

Methods to Find Vertical Asymptotes

There are mainly 2 ways of finding vertical asymptote:

1. Vertical Asymptotes From Graph

Vertical asymptotes are vertical lines that a function's graph approaches but never actually reaches or crosses. They occur where the function is undefined or approaches infinity.

Step 1: Look for vertical traces that the graph approaches however in no way touches.

Step 2: The graph will typically approach positive infinity on one side and negative infinity on the other side of the asymptote.

Step 3: There is often a break or gap in the graph at the asymptote.

2. Vertical Asymptotes From Equation

According to definition of Vertical Asymptotes, a vertical asymptote occurs at x = k for a function f(x) if:

• lim_x→k f(x) = ∞, or
• lim_x→k f(x) = -∞

To find Vertical Asymptotes, look for x-values that make the limit of the function approach infinity (positive or negative). Let us understand it through examples.

Example 1: For f(x) = 1/(x+1).

Vertical Asymptote at x = -1

Because lim_{x → -1} 1/(x+1) = ∞

Vertical Asymptotes of Rational Function

Vertical asymptotes are vertical strains that a function's graph tactics however in no way honestly reaches. For rational capabilities, these arise wherein the denominator equals 0, and the numerator is not zero.

Steps to find Vertical Asymptotes:

Step 1 : Express the rational feature inside the form f(x) = P(x) / Q(x), in which P(x) and Q(x) are polynomials.

Step 2 : Find the values of x that make Q(x) = 0.

Step 3 : Check if those x values moreover make P(x) = 0. If not, they represent vertical asymptotes.

Example: Find the vertical asymptotes of f(x) = 1 / (x - 2).

Step 1: The function is already in the form P(x) / Q(x).

P(x) = 1, Q(x) = x - 2

Step 2: Solve Q(x) = 0

x - 2 = 0

⇒ x = 2

Step 3: Check if P(2) = 0

P(2) = 1 ≠ 0

Therefore, x = 2 is a vertical asymptote.

Vertical Asymptotes of Trigonometric Functions

The vertical asymptotes of trigonometric functions are presented in the tabular form below:

To determine the vertical asymptotes for these functions, consider the values of x that make each function undefined. These are the points where the function approaches infinity or negative infinity.

Example: Find vertical asymptotes of f(x)=tan⁡x

\tan x = \frac{\sin x}{\cos x}

Asymptote occurs when denominator = 0:

cos⁡x=0

x=\frac{\pi}{2} + n\pi

So, Vertical asymptotes: x=\frac{\pi}{2} + n\pi

Vertical Asymptote of Logarithmic Function

For a logarithmic feature of the form f(x) = log_b(x - h) ok, where b is the base, the vertical asymptote happens where the argument of the logarithm equals 0.

f(x)= a ⋅ log_b​(x−h)

Vertical Asymptote: x = h

Key Points:

• The vertical asymptote is always on the left side of the graph for logarithmic functions.
• The function is undefined for all x values less than h.
• As x approaches h from the right, the function value approaches negative infinity.

Example : Find the vertical asymptote of f(x) = log₂(x + 3)

Here, we have h = -3 (the argument is shifted 3 units left).

The vertical asymptote is at x = -3.

Vertical Asymptotes of Exponential Function

The general form of an exponential function is f(x) = a^x, where a is a positive constant and x is the variable.

• Domain: The area of an exponential function is all real numbers (x ∈ R). This method that for any actual range input, the characteristic will produce a legitimate output.
• No vertical asymptotes: Since the characteristic is described for all real numbers, there's no x-price where the feature becomes undefined or techniques infinity. Therefore, exponential functions do not have vertical asymptotes.
• Horizontal asymptote: While there's no vertical asymptote, exponential functions do have a horizontal asymptote at y = zero as x methods negative infinity. This is due to the fact lim(x→-∞) a^x = zero for a > 1.

Example: Determine whether the function f(x) = \frac{e^x}{x - 1} has a vertical asymptote.

Step 1: Identify where function is undefined

The denominator becomes zero when: x−1=0 ⇒ x=1

Step 2: Check behavior near x=1

Evaluate limits:\lim\limits_{x \to 1^-} \frac{e^x}{x-1} = -\infty ,\lim\limits_{x \to 1^+} \frac{e^x}{x-1} = +\infty

• As x→1⁻: denominator is negative so function → −∞
• As x→1⁺: denominator is positive so function → +∞

Since the function approaches ±∞ near x=1 so vertical asymptote: x=1

Solved Problems

Problem 1: Find the vertical asymptote of f(x) = 1 / (x + 2)

Set denominator to zero: x + 2 = 0

Solve for x: x = -2

Vertical asymptote: x = -2

Problem 2: Determine the vertical asymptote of g(x) = (x² - 1) / (x - 1)

Set denominator to zero: x - 1 = 0

Solve for x: x = 1

Check if numerator is also zero when x = 1: 1² - 1 = 0

Since numerator is also zero, simplify function:

g(x) = (x + 1)(x - 1) / (x - 1) = x + 1

There is no vertical asymptote.

Problem 3: Find the vertical asymptote of h(x) = 1 / √(4 - x²)

Set expression under square root to zero: 4 - x² = 0

Solve for x: x² = 4, x = ±2

Vertical asymptotes: x = 2 and x = -2

Problem 4: Find the vertical asymptote of n(x) = ln(x - 5)

The argument of ln must be positive: x - 5 > 0

Solve for x: x > 5

Vertical asymptote: x = 5

Problem 5: Identify the vertical asymptote of p(x) = (x³ - 8) / (x - 2)

Set denominator to zero: x - 2 = 0

Solve for x: x = 2

Check if numerator is also zero when x = 2: 2³ - 8 = 0

Since numerator is also zero, simplify function:

p(x) = x² + 2x + 4

There is no vertical asymptote.

Problem 6: Identify the vertical asymptote of s(x) = e^x / (x - 3)

Set denominator to zero: x - 3 = 0

Solve for x: x = 3

Vertical asymptote: x = 3

Practice Questions

Q1: Find the vertical asymptote(s) of f(x) = 1 / (x - 3)

Q2: Determine the vertical asymptote(s) of g(x) = (x² - 1)/(x² - 4)

Q3: Identify the vertical asymptote(s) of h(x) = (x - 2) / (x² - x - 6)

Q4: Find the vertical asymptote(s) of f(x) = √[(x - 2)/(x - 1)]

Q5: Determine the vertical asymptote(s) of m(x) = (x³ - 8) / (x² - 4)

Related Articles

• How to find Vertical and Horizontal Asymptotes?
• Asymptote Formula