When is there horizontal asymptote

The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator. Both the numerator and denominator are linear degree 1. Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is t , with coefficient 1.

In the denominator, the leading term is 10 t , with coefficient The horizontal asymptote will be at the ratio of these values:. First, note that this function has no common factors, so there are no potential removable discontinuities. The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined.

The numerator has degree 2, while the denominator has degree 3. A rational function will have a y -intercept when the input is zero, if the function is defined at zero. A rational function will not have a y -intercept if the function is not defined at zero.

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. In calculus, you'll learn how to prove this yourself.

In the example above, the degrees on the numerator and denominator were the same, and the horizontal asymptote turned out to be the horizontal line whose y -value was equal to the value found by dividing the leading coefficients of the two polynomials.

This is always true: When the degrees of the numerator and the denominator are the same, then the horizontal asymptote is found by dividing the leading terms, so the asymptote is given by:.

Now that I know the rules about the powers, I don't have to do a table of values or draw the graph. I can just compare exponents. In this rational function, the highest power in each of the numerator and the denominator is the same; namely, the cube. 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. Lesson: 2b. Lesson: 2c. Lesson: 3. Lesson: 3a. Lesson: 3b.

Lesson: 3c.