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A vertical asymptote is a vertical line that seems to coincide with the graph of a function but it actually never meet the curve. A vertical asymptote of a function plays an important role while graphing a function.
Let us learn more about the vertical asymptote along with the process of finding it for different types of functions.
What is Vertical Asymptote?
The vertical asymptote of a function y = f(x) is a vertical line x = k when y→∞ or y→ -∞. It is usually referred to as VA. Mathematically, if x = k is the VA of a function y = f(x) then atleast one of the following would hold true:
- lim x→k f(x) = ±∞ (or)
- lim x→k₊ f(x) = ±∞ (or)
- lim x→k- f(x) = ±∞
In other words, at vertical asymptote, either the left-hand side (or) the right-hand side limit of the function would be either ∞ or -∞.
A vertical asymptote is a vertical line along which the function becomes unbounded (either y tends to ∞ or -∞) but it doesn't touch or cross the curve. If x = k is the VA of a function y = f(x) then k is NOT present in the domain of the function. A function can have any number of vertical asymptotes. i.e., it can have 0, 1, 2, ..., or an infinite number of VAs. We represent a VA by a vertical dotted line and if the y-axis is the VA, then we usually do not show it by a dotted line. Here are a few examples of vertical asymptotes.
The graph of a function can never cross the VA and hence it is NOT a part of the curve anymore. We find vertical asymptotes while graphing but it is not mandatory to show them on the graph. Even the graphing calculators do not show them explicitly with dotted lines.
How to Find Vertical Asymptotes?
Vertical asymptotes can be determined from the graphs and as well as the equations of functions. We will learn about each of the cases.
Vertical Asymptotes From Graph
By seeing the above examples, you might have already got an idea of determining the vertical asymptotes from a graph. If a part of the graph is turning to be vertical, then there might probably be a VA along that vertical line. The value of the function becomes ∞ or -∞ at the value of x along which you found the VA. But note that a vertical asymptote should never touch the graph.
Vertical Asymptotes From Equation
From the definition of vertical asymptote, if x = k is the VA of a function f(x) then lim x→k f(x) = ∞ (or) lim x→k f(x) = -∞. To identify them, just think what values of x would make the limit of the function to be ∞ or -∞. Observe the above graphs
- VA of f(x) = 1/(x+1) is x = -1 as lim x → -1 1/(x+1) = ∞.
- VAs of f(x) = 1/[(x+1)(x-2)] are x = -1 and x = 2 as the left/right hand limits at each of x = -1 and x = 2 is either ∞ or -∞.
To know how to evaluate the limits, click here. Let us see how to find the vertical asymptotes of different types of functions using some tricks/shortcuts.
Vertical Asymptotes of Rational Function
We do not need to use the concept of limits (which is a little difficult) to find the vertical asymptotes of a rational function. Instead, use the following steps:
- Step 1: Simplify the rational function. i.e., Factor the numerator and denominator of the rational function and cancel the common factors.
- Step 2: Set the denominator of the simplified rational function to zero and solve.
Here is an example to find the vertical asymptotes of a rational function.
Example: Find vertical asymptotes of f(x) = (x + 1) / (x2 - 1).
Let us factorize and simplify the given expression:
Then f(x) = (x + 1) / [ (x + 1) (x - 1) ] = 1 / (x - 1).
Now, set the denominator to zero. Then
(x - 1) = 0
x = 1
So x = 1 is the VA of f(x).
Note that do not set the denominator = 0 directly without simplifying the function. If we do that, we get x = -1 and x = 1 to be the VAs of f(x) in the above example. But x = -1 is NOT a VA anymore in this case, because (x + 1) has got canceled while simplification. In fact, there will be a hole at x = -1. Here is the graph of the function to understand the difference between the vertical asymptotes and holes.
Vertical Asymptotes of Trigonometric Functions
Among the 6 trigonometric functions, 2 functions (sine and cosine) do NOT have any vertical asymptotes. But each of the other 4 trigonometric functions (tan, csc, sec, cot) have vertical asymptotes. To find them, just think about what values of x make the function undefined. Here are the vertical asymptotes of trigonometric functions:
- y = sin x has no vertical asymptotes.
- y = cos x has no vertical asymptotes.
- The vertical asymptotes of y = tan x are at x = πn + π/2, where 'n' is an integer.
- The vertical asymptotes of y = csc x are at x = πn, where 'n' is an integer.
- The vertical asymptotes of y = sec x are at x = πn + 3π/2, where 'n' is an integer.
- The vertical asymptotes of y = cot x are at x = πn, where 'n' is an integer.
You can see the graphs of the trigonometric function by clicking here and you can observe the VAs of all trigonometric functions in the graphs.
Vertical Asymptote of Logarithmic Function
We know that the value of a logarithmic function f(x) = loga x or f(x) = ln x becomes unbounded when x = 0. So the vertical asymptote of a basic logarithmic function f(x) = loga x is x = 0. We can observe this in the graph below.
So the vertical asymptote of any logarithmic function is obtained by setting its argument to zero. Here are more examples:
- VA of f(x) = log (x + 1) is x + 1 = 0 ⇒ x = -1.
- VA of f(x) = ln (x - 2) is x - 2 = 0 ⇒ x = 2.
Vertical Asymptotes of Exponential Function
The parent exponential function is of the form f(x) = ax and after transformations, it may look like f(x) = bacx + k. Do you think the exponential function goes undefined for any value of x? No, an exponential function is defined for all real values of x and hence it has no vertical asymptotes. Here is an example.
Vertical Asymptote Rules
Let us summarize the rules of finding vertical asymptotes all at one place:
- To find the vertical asymptotes of a rational function, simplify it and set its denominator to zero.
- Exponential functions and polynomial functions (like linear functions, quadratic functions, cubic functions, etc) have no vertical asymptotes.
- To find the vertical asymptotes of logarithmic function f(x) = log (ax + b), set ax + b = 0 and solve for x.
- All trigonometric functions except sin x and cos x have vertical asymptotes. The VAs of
tan x are x = πn + π/2
csc x are x = πn
sec x are x = πn + 3π/2
cot x are x = πn
where, n is an integer.
- To find the vertical asymptote of any other function than these, just think what values of x would make the function to be ∞ or -∞.
Important Notes on Vertical Asymptotes:
- A function can have any number of vertical asymptotes.
- No polynomial function has a vertical asymptote.
- No exponential function has a vertical asymptote.
- Every logarithmic function has at least one vertical asymptote.
- Simplify the rational functions first before setting the denominator to 0 while finding the vertical asymptotes.
☛ Related Topics:
Vertical Asymptote Examples
Example 1: Find vertical asymptote of f(x) = (3x2)/(x2-5x+6).
The given function is a rational function. To find its VA, we need to simplify it first.
It is already in the simplest form. So we set the denominator = 0 and solve for x values.
x2-5x+6 = 0
Factoring this quadratic expression,
(x - 2) (x - 3) = 0
x - 2 = 0, x - 3 = 0
x = 2, x = 3
Answer: VAs of the function are x = 2 and x = 3.
Example 2: Find vertical asymptote(s) of f(x) = (x2 - 2x) / (x - 2).
Let us simplify the function first by factoring.
f(x) = [x (x - 2)] / (x - 2) = x
We know that f(x) = x is a linear function and hence it has no vertical asymptotes.
Answer: The given function has no VA but it has a hole at x = 2.
Example 3: The vertical asymptote of a function f(x) = log (2x - k) is x = 3. Then what is k?
The VA of the given function is obtained by setting 2x - k = 0.
Solving this, we get 2x = k (or) x = k/2.
But it is given that VA is x = 3.
So k/2 = 3. From this, k = 6.
Answer: k = 6.
Practice Questions on Vertical Asymptote
FAQs on Vertical Asymptote
How to Find Vertical Asymptote of a Function?
The vertical asymptote is a type of asymptote of a function y = f(x) and it is of the form x = k where the function is not defined at x = k. i.e., the left hand/right hand/ both limits of the function is either equal to ∞ or -∞ as x tends to k.
How to Find Vertical Asymptote From a Graph?
To find the vertical asymptote from the graph of a function, just find some vertical line to which a portion of the curve is parallel and very close. It is of the form x = k. Remember that as x tends to k, the limit of the function should be an undefined value. i.e., the graph should continuously extend either upwards or downwards.
What are the Vertical Asymptotes of Trigonometric Functions?
All trigonometric functions do not have vertical asymptotes (VAs). Only tan, csc, sec, and cot have them.
- VAs of y = tan x are x = πn + π/2
- VAs of y = csc x are x = πn
- VAs of y = sec x are x = πn + 3π/2
- VAs of y = cot x are x = πn
Here, n is an integer.
How to Find Vertical Asymptote of a Rational Function?
To find the vertical asymptotes of a rational function, just get the function to its simplest form, set the denominator of the resultant expression to zero, and solve for x values.
What Functions Do Not have Vertical Asymptotes?
Polynomial functions like linear, quadratic, cubic, etc; the trigonometric functions sin and cos; and all the exponential functions do NOT have vertical asymptotes.
What is the Difference Between Vertical Asymptote and Horizontal Asymptote?
Asymptote (vertical/horizontal) is an imaginary line to which a part of the curve seems to be parallel and very close. A horizontal asymptote is a horizontal line and is in the form y = k and a vertical asymptote is a vertical line and is of the form x = k, where k is a real number.