How do horizontal asymptotes work

Lesson: 1b. Lesson: 1c. Lesson: 2. Lesson: 2a. Lesson: 2b. Lesson: 2c. Lesson: 3. Lesson: 3a. Lesson: 3b. Lesson: 3c. Lesson: 3d. Lesson: 3e.

Lesson: 3f. Lesson: 3g. Lesson: 3h. 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. Likewise, a rational function will have x -intercepts at the inputs that cause the output to be zero.

Then, find the x — and y -intercepts and the horizontal and vertical asymptotes. In other words, this rational function has no vertical asymptotes. So we're okay on that front. As mentioned above, the horizontal asymptote of a function assuming it has one tells me roughly where the graph will being going when x gets really, really big.

So I'll look at some very big values for x ; that is, at some values of x which are very far from the origin:. I ended up having a really big number divided by a really big number squared, which "simplified" to be a very small number. The values of y came mostly from the " x " and the " x 2 ", especially once x got very large. This makes perfect sense, when you think about it.

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:.