The asymptotes of a function are values that a function approaches as the values of x approach a specific value. For example, a function can approach but never reach, the *x*-axis as the *x* values tend to infinity.

Here, we will look at a summary of the three types of asymptotes that functions can have. In addition, we will look at several examples with answers of asymptotes in order to learn how to find the asymptotes of functions.

## Summary and examples of vertical asymptotes

To find the vertical asymptotes of a function, we have to examine the factors of the denominator that are not common with the factors of the numerator. The zeros of these factors represent the vertical asymptotes.

We can use the following steps to identify the vertical asymptotes of rational functions:

**Step 1:** If possible, factor the numerator and denominator.

** Step 2:** Determine if the domain of the function has any restrictions.

** Step 3:** Cancel common factors if any to simplify to the expression.

** Step 4:** If there is a value in the simplified version that makes the denominator zero, then those values represent the vertical asymptotes.

### EXAMPLE 1

Considering the rational function $latex f(x)= \frac{{{x}^2}+2x-3}{{{x}^2}-5x-6}$, find its vertical asymptotes.

**Solution:** We can factor both the numerator and the denominator as follows:

$latex f(x)= \frac{(x+3)(x-1)}{(x-6)(x+1)}$

Looking at the denominator, we know that *x* cannot be either 6 or -1 since we would have division by zero. The following is the graph of the rational function:

In the graph we see that the curves avoid the vertical lines $latex x = 6$ and $latex x = -1$. The values of *x* cannot be equal to 6 or -1, so these are the asymptotes. We can draw the vertical asymptotes as dashed lines:

### EXAMPLE 2

Given the function $latex g(x)= \frac{x+2}{{{x}^2}+2x-8}$, find its vertical asymptotes.

**Solution:** Here, we just have to factor the denominator:

$latex f(x)= \frac{x+2}{(x+4)(x-2)}$

Looking at the denominator, we know that *x* cannot be equal to $latex x=-4$ or $latex x =2 $ as this would cause division by zero. This tells us that the vertical asymptotes of the function are located at $latex x=-4$ and $latex x=2$:

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## Summary and examples of horizontal asymptotes

To find the horizontal asymptotes of rational functions, we can use the following methods that vary depending on how the degrees of the polynomial compare in the numerator and denominator of the function:

**•** When the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator, we divide the leading coefficients (of the variable with the highest exponent) to obtain the horizontal asymptotes.

**• **When the degree of the numerator is less than the degree of the denominator, we have the horizontal asymptote $latex y=0$.

**• **When the degree of the numerator is greater than the degree of the denominator, then the function has no horizontal asymptotes.

Let’s look at some exercises:

### EXAMPLE 1

Given the function $latex g(x)=\frac{x+2}{2x}$, determine its horizontal asymptotes.

**Solution:** In both the numerator and the denominator, we have a polynomial of degree 1. Therefore, we find the horizontal asymptote by considering the coefficients of *x*.

Thus, the horizontal asymptote of the function is $latex y=\frac{1}{2}$:

### EXAMPLE 2

Given the function $latex g(x)=\frac{x}{{{x}^2}+2}$, determine its horizontal asymptotes.

**Solution:** In this function, the degree of the numerator is less than the degree of the denominator. This means that the horizontal asymptote is located at $latex y = 0$:

### EXAMPLE 3

Given the function $latex f(x)=\frac{{{x}^2}+2}{x+1}$, find its horizontal asymptotes.

**Solution:** We see that the degree of the numerator is greater than the degree of the denominator. This means that the function does not have a horizontal asymptote:

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## Summary and examples of oblique asymptotes

Oblique asymptotes, also called slanted, can be determined by comparing the degree of the numerator and the degree of the denominator.

When the degree of the numerator is exactly one more than the degree of the denominator, then the rational function will produce a graph that will look roughly like an inclined line with complicated divergences in the middle. The asymptote of this type of function is called an oblique or slanted asymptote.

To obtain the equation of this asymptote, we have to perform long division of polynomials. The equation of the asymptote is the integer part of the result of the division.

### EXAMPLE

Given the function $latex f(x)=\frac{-3{{x}^2}+2}{x-1}$, determine its oblique asymptote.

**Solution:** We have to perform the long division of this rational function:

The quotient of the division, located at the top, is the linear expression $latex -3x-3$. The residue is located at the bottom. Therefore after division, we convert the function into a mixed expression:

$latex f(x)=-3x-3+\frac{-1}{x-1}$

The oblique asymptote of the function is the line $latex y=-3x-3$, that is, the integer part of the division:

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## See also

Interested in learning more about algebraic topics? Take a look at these pages:

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