Is it possible to have a horizontal and oblique asymptote

Main Concept. An asymptote is a line that the graph of a function approaches as either x or y go to positive or negative infinity. There are three types of asymptotes: vertical, horizontal and oblique. Vertical Asymptotes. Vertical Asymptote. That is, as x approaches a from either the positive or negative side, the function approaches positive or negative infinity. Vertical asymptotes occur at the values where a rational function has a denominator of zero.

The function is undefined at these points because division by zero mathematically ill-defined. Horizontal Asymptotes. Horizontal Asymptote. This means, that as x approaches positive or negative infinity, the function tends to a constant value a. Horizontal asymptotes occur when the numerator of a rational function has degree less than or equal to the degree of the denominator. If you do, then I'll be a Hersey bar that the graph violates the vertical line test, and therefor is not the graph of a function.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why can't a rational function have both, a horizontal and an oblique asymptote?

Ask Question. Asked 5 years, 1 month ago. Active 5 years, 1 month ago. Viewed 13k times. Allen Allen 33 1 1 gold badge 1 1 silver badge 3 3 bronze badges. Non-rational functions can. Add a comment. Active Oldest Votes. Note that vertical asymptotes, i. Learning Objectives After completing this tutorial, you should be able to: Find the domain of a rational function. Find the vertical asymptote s of a rational function. Find the horizontal asymptote of a rational function.

Find the oblique or slant asymptote of a rational function. Graph a rational function. Introduction In this tutorial we will be looking at several aspects of rational functions.

First we will revisit the concept of domain. On rational functions, we need to be careful that we don't use values of x that cause our denominator to be zero. If you need a review on domain, feel free to go to Tutorial Introductions to Functions.

Next, we look at vertical, horizontal and slant asymptotes. Basically an asymptote is an imaginary line that the curve of the function gets very close to or approaches. In the end, we put it all together and graph rational functions. Sounds like fun, you better get to it!!! Tutorial Review on Domain The domain is the set of all input values to which the rule applies. These are called your independent variables. These are the values that correspond to the first components of the ordered pairs it is associated with.

Example 1 : Give the domain of the function. Our restriction here is that the denominator of a fraction can never be equal to 0. So to find our domain, we want to set the denominator equal to 0 and restrict those values. Vertical Asymptote Let be written in lowest terms and P and Q are polynomial functions. This is where the function is undefined, so there will be NO point on the vertical asymptote itself.

The graph will approach it from both sides, but never cross over it. First we want to check and see if this rational function will reduce down :. Let be written in lowest terms, where P and Q are polynomial functions and. The slant asymptote is the quotient part of the answer you get when you divide the numerator by the denominator. If you need a review of long division, feel free to go to Tutorial Long Division.

Note that this rational function is already reduced down. Applying long division to this problem we get:. Practice Problems. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problem 1a: Give the domain of the given function. Practice Problems 2a - 2b: Find the vertical and horizontal asymptotes for the given functions. Practice Problem 3a: Find the oblique asymptote for the given function.

Practice Problems 4a - 4b: Sketch the graph of the given function. Need Extra Help on these Topics? The following are webpages that can assist you in the topics that were covered on this page. All rights reserved. After completing this tutorial, you should be able to: Find the domain of a rational function. In this tutorial we will be looking at several aspects of rational functions.

The domain is the set of all input values to which the rule applies. In other words, you find the vertical asymptote by locating where the function is undefined. You can have zero or many vertical asymptotes. Example 2 : Find the vertical asymptote of the function.