When you're working with rational functions and their asymptotes, it’s easy to make some common mistakes. Here are some tips to help you understand and analyze these functions better.

1. Misidentifying Vertical Asymptotes

A vertical asymptote happens where the denominator equals zero. This is where the function isn’t defined.

Don’t mix this up with x-values where the function just goes down.

For example, in the function:

$f(x) = \frac{1}{x - 2}$

the vertical asymptote is at $x = 2$ because that’s where the denominator becomes zero.

Make sure you don’t miss or mess up any factors in the denominator.

2. Forgetting to Simplify

Always simplify your rational function before you look for asymptotes.

For example:

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

can be simplified to:

$f(x) = \frac{(x - 1)(x + 1)}{x + 1}$

This shows there’s a hole in the graph at $x = -1$ instead of a vertical asymptote.

If you forget to simplify, you might mistakenly think there is a vertical asymptote at that point.

3. Confusing Horizontal and Vertical Asymptotes

Horizontal asymptotes show how the function behaves as $x$ gets really big or really small.

They are based on the degrees of the polynomial in the top (numerator) and bottom (denominator).

For example:

• If the degree of the numerator is less than the degree of the denominator, like in

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

then the horizontal asymptote is $y = 0$.

• If the degrees are the same, you look at the leading numbers (coefficients) to find the horizontal asymptote.

4. Ignoring End Behavior

Sometimes, students forget to think about how the function acts as $x$ goes towards positive or negative infinity.

This is really important for making a good graph.

For instance, with the function:

$f(x) = \frac{2x^3 + 3}{x^3 - 5}$

As $x$ approaches infinity, the horizontal asymptote is $y = 2$. This tells you the end behavior of the function.

By keeping these common mistakes in mind, you’ll be on your way to mastering rational functions and their asymptotes!