Vertical Asymptote

Vertical asymptotes, or VA, are dashed vertical lines on a graph corresponding to the zeroes of a function y = f(x) denominator. Thus, the curve approaches but never crosses the vertical asymptote, as that would imply division by zero.

We get the VA of the function as x = c when x approaches a constant value c going from left to right, and the graph tends to infinity or –infinity, as shown below. Note that c is not part of the domain of f(x). Hence, we can say that the vertical asymptotes of a function are not part of its domain.

A function can have infinite vertical asymptotes, which can be negative and extend to infinity. Unlike horizontal asymptotes, the curve never crosses the vertical asymptote.

How to Find Vertical Asymptotes

We can determine the VA of a function f(x) from its graph or equation.

From a Graph

When looking at the f(x) graph, if any parts appear to be vertical, they are probably vertical asymptotes. We can check by drawing a vertical line from the point that looks like a VA. The curve is not a VA if it touches any part of this line.

From an Equation

Without a graph, we can find the vertical asymptotes from the equation of the function itself. To do so, we need to find the values of x such that ${ \lim _{x\rightarrow c}f\left( x\right) \rightarrow \infty}$ or ${\lim _{x\rightarrow c}f\left( x\right) \rightarrow -\infty}$. The values of c obtained from solving the above will give us all the vertical asymptotes.

While this rule gives us all the vertical asymptotes for most functions, there are a few more straightforward methods for some cases, which we will discuss below.

For Rational Functions

For rational functions of the form ${f\left( x\right) =\dfrac{P\left( x\right) }{Q\left( x\right) }}$, where P(x) and Q(x) are polynomials, we can calculate the VA using these rules:

• First, factorize the numerator and denominator, respectively, of the rational function.
• Next, simplify by canceling the common terms between the numerator and denominator. This step is important as these terms do not give a vertical asymptote.
• Finally, set the simplified denominator to zero and solve the remaining terms for x. The zeroes of the denominator are thus the vertical asymptotes of the function f(x).

For example, let us take the function ${f\left( x\right) =\dfrac{x^{2}+4x+4}{x^{2}+x-2}}$.

On factoring the numerator and denominator, we get

${f\left( x\right) =\dfrac{\left( x+2\right) \left( x+2\right) }{\left( x-1\right) \left( x+2\right) }}$

${\Rightarrow f\left( x\right) =\dfrac{\left( x+2\right) }{\left( x-1\right) }}$

Thus, (x – 1) remains the denominator after removing the common terms. Solving this for zero, we get x = -1 as the vertical asymptote for f(x).

The common factor (x + 2) in the above example is a removable discontinuity or a hole. It is a point where the graph is undefined, indicated by an open circle. For f(x), the point (0,-2) is the removable discontinuity.

For Trigonometric Functions

Of the six basic trigonometric functions, sine and cosine do not have any VA. The other four, namely tan, sec, cosec, and cot, all have vertical asymptotes for the following values of x.

• y = tan x has vertical asymptotes at ${x=n\pi +\dfrac{\pi }{2}}$, where ‘n’ is an integer.
• y = sec x has vertical asymptotes at ${x=n\pi +\dfrac{\pi }{2}}$, where ‘n’ is an integer.
• y = cosec x has vertical asymptotes at x = nπ, where ‘n’ is an integer.
• y = cot x has vertical asymptotes at x = nπ, where ‘n’ is an integer.

For Logarithmic Functions

The vertical asymptote of any logarithmic function is found by setting its argument to zero and solving for x. From this rule, the VA of a standard logarithmic function ${ f\left( x\right) =\log _{a}x}$ or f(x) = ln x is x = 0.

For Other Functions and Exceptions

1. Rectangular hyperbolic functions of the form ${ f\left( x\right) =\dfrac{a}{x+p}+q}$ have a vertical asymptote at x = -p.
2. Polynomial functions of the form ${f\left( x\right) =a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x^{1}+a_{0}}$ do not have any vertical asymptote as x does not give any real values at limits of infinity and –infinity.
3. Exponential functions of the form ${ f\left( x\right) =a^{x}}$ or ${f\left( x\right) =ba^{cx}+k}$ are defined for all real values of x and so have no vertical asymptotes.

Solved Examples

Now we have the only remaining term in the denominator as (x + 4). Putting it as zero and solving for x, we get x = -4 as the vertical asymptote of the function f(x).