When dealing with rational functions and asymptotes, students can sometimes make a few common mistakes. These mistakes can cause confusion and lead to errors in their calculations. Let's look at these mistakes and how to avoid them.

1. Forgetting to Factor:
One frequent mistake is not fully factoring both the top and bottom of the fraction.

For example, take this function:
$f(x) = \frac{x^2 - 1}{x^2 - 4}$
Before finding asymptotes, it’s crucial to factor both parts:
$f(x) = \frac{(x - 1)(x + 1)}{(x - 2)(x + 2)}$

If you skip this step, you might come to the wrong conclusions about vertical asymptotes and holes in the graph.

2. Misidentifying Vertical Asymptotes:
Sometimes, students mistakenly think a function has a vertical asymptote at any point where the bottom (denominator) is zero.

However, it’s important to check for common factors in the top (numerator).

Using our example above, the function has vertical asymptotes at $x = 2$ and $x = -2$. But there is also a removable discontinuity (a hole) at $x = 1$. That’s because both the top and bottom go to zero at that point. This is important to fully understand how the function behaves.

3. Overlooking Horizontal Asymptotes:
Horizontal asymptotes can also confuse students. Sometimes they ignore them or make mistakes when calculating their values.

Here’s a simple rule: for rational functions, if the degree (the highest power) of the top is less than that of the bottom, the horizontal asymptote is at $y = 0$.

If the degrees are the same, the asymptote is at $y = \frac{a}{b}$, where $a$ and $b$ are the numbers in front of the highest powers.

For instance, with the function
$f(x) = \frac{3x^2 + 1}{2x^2 + 5},$
the horizontal asymptote is at $y = \frac{3}{2}$ since both degrees are 2.

4. Confusing Asymptotes with the Graph’s Path:
Lastly, students often misunderstand what asymptotes are. Remember, asymptotes are lines that the graph gets closer to but doesn’t actually touch.

For example, if $f(x)$ approaches $y = 0$, it doesn’t mean the graph will actually hit that line.

By avoiding these common mistakes, you’ll have a better understanding of rational functions and how they behave. This will help you do better in your math classes!