When is horizontal asymptote 1

Skip to main content. Rational Functions. Search for:. Identify horizontal asymptotes While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. A General Note: Horizontal Asymptotes of Rational Functions The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.

Degree of numerator is greater than degree of denominator by one : no horizontal asymptote; slant asymptote. Degree of numerator is equal to degree of denominator: horizontal asymptote at ratio of leading coefficients. Example 7: Identifying Horizontal and Slant Asymptotes For the functions below, identify the horizontal or slant asymptote.

Find the horizontal asymptote and interpret it in context of the problem. Solution Both the numerator and denominator are linear degree 1.

Solution First, note that this function has no common factors, so there are no potential removable discontinuities.

If you've got a zillion plus two, but who cares about that? Which is very, very small. So of course the value of the function gets very, very small; namely, it gets very, very close to zero.

I can see this behavior on the graph, if I zoom out on the x -axis:. The graph shows that there's some slightly interesting behavior in the middle, right near the origin, but the rest of the graph is fairly boring, trailing along the x -axis.

If I zoom in on the origin, I can also see that the graph crosses the horizontal asymptote at the arrow :. It is common and perfectly okay to cross a horizontal asymptote.

It's the vertical asymptotes that I'm not allowed to touch. As I can see in the table of values and the graph, the horizontal asymptote is the x -axis. This property is always true: If the degree on x in the denominator is larger than the degree on x in the numerator, then the denominator, being "stronger", pulls the fraction down to the x -axis when x gets big.

That is, if the polynomial in the denominator has a bigger leading exponent than the polynomial in the numerator, then the graph trails along the x -axis at the far right and the far left of the graph. What happens if the degrees are the same in the numerator and denominator?

Let's take a look:. Unlike the previous example, this function has degree- 2 polynomials top and bottom; in particular, the degrees are the same in the numerator and the denominator. Since the degrees are the same, the numerator and denominator "pull" evenly; this graph should not drag down to the x -axis, nor should it shoot off to infinity. But where will it go? Again, I need to think in terms of big values for x.

When x is really big, I'll have, roughly, twice something big minus an eleven, but who cares about that? And the graph of the function reflects this:. Sure, there's probably something interesting going on in the middle of the graph, near the origin. You can still navigate around the site and check out our free content, but some functionality, such as sign up, will not work.

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That's the lesson. That's the last lesson. Let's keep going. Play next lesson or Practice this topic. Start now and get better math marks! Intro Lesson. Lesson: 1. Lesson: 1a. Lesson: 1b. Lesson: 1c. Lesson: 2. Lesson: 2a.