When do you have oblique asymptotes

If the degree of the numerator top is exactly one greater than the degree of the denominator bottom , then f x will have an oblique asymptote. So there are no oblique asymptotes for the rational function,. But a rational function like does have one. Knowing when there is a horizontal asymptote is just half the battle. Now how do we find it? This next step involves polynomial division. Another place where oblique asymptotes show up is in the graphs of hyperbolas.

Remember, in the simplest case, a hyperbola is characterized by the standard equation,. Furthermore, if the center of the hyperbola is at a different point than the origin, h , k , then that affects the asymptotes as well.

Below is a summary of the various possibilities. Keeping these techniques in mind, oblique asymptotes will start to seem much less mysterious on the AP exam!

In addition, Shaun earned a B. Shaun still loves music -- almost as much as math! Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed! View all posts.

Click here to learn more! 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 the denominator has degree n , the horizontal asymptote can be calculated by dividing the coefficient of the x n -th term of the numerator it may be zero if the numerator has a smaller degree by the coefficient of the x n -th term of the denominator. Oblique Asymptotes. Oblique Asymptote. Oblique asymptotes occur when the degree of the denominator of a rational function is one less than the degree of the numerator.

But what happens if the degree is greater in the numerator than in the denominator? Recall that, when the degree of the denominator was bigger than that of the numerator, we saw that the value in the denominator got so much bigger, so quickly, that it was so much "stronger" that it "pulled" the functional value down to zero, giving us a horizontal asymptote of the x -axis.

To investigate this , let's look at the following function:. For reasons that will shortly become clear, I'm going to apply long polynomial division to this rational expression. My work looks like this:. Across the top is the quotient, being the linear polynomial expression —3 x — 3. At the bottom is the remainder. This means that, via long division, I can convert the original rational function they gave me into something akin to mixed-number format:.

This is the exact same function. All I've done is rearrange it a bit. You're about to see. Clearly, it's not a horizontal asymptote.

Instead, because its line is slanted or, in fancy terminology, "oblique", this is called a "slant" or "oblique" asymptote.