Finding oblique asymptotes in rational functions can be made easier with a few simple steps. Let’s go through them!

1. Look at the Degrees:

First, check the degrees of the top and bottom parts of the fraction (called polynomials).

• If the degree of the top (numerator) is one more than the degree of the bottom (denominator), then you will have an oblique asymptote.

2. Use Long Division:

When you see that degree difference, you can do polynomial long division. Here’s how it works:

• Divide the top polynomial by the bottom polynomial.
• The result will be a straight line (this is your oblique asymptote) plus a small leftover part (which gets smaller as $x$ gets really big or really small).

For example, let’s look at this function:

$f(x) = \frac{2x^3 + 3x^2 + 1}{x^2 + 1}.$

Here, the degree of the top (3) is one more than the degree of the bottom (2). So, you would do long division in this case.

3. Forget the Leftover Part:

After you finish the long division, you can ignore that small leftover part when you find the oblique asymptote. Just focus on the straight line you got from the division.

4. Draw It:

Finally, to really understand what you’re seeing, draw the original function next to the oblique asymptote. This will help you see how the function behaves as $x$ gets really large or really small.

By using these steps, you can easily find oblique asymptotes and understand how rational functions act at the ends. It’s pretty cool to see these ideas come to life when you graph them!