Rational functions can be tricky, especially when we're trying to understand what happens near asymptotes.

1. Types of Asymptotes:

   • Vertical Asymptotes: These happen where the function isn't defined. At these points, the value of the function can change a lot, really fast.
   • Horizontal Asymptotes: These show how the function behaves as we move towards the far ends of the graph. However, they can be a bit confusing for values in between.

2. Difficulties:

   • When you get close to a vertical asymptote, the function might explode towards positive or negative infinity. This makes it hard to understand what the limits are.
   • Figuring out what exactly happens near the asymptote can mean doing a lot of complicated math.

3. Solutions:

   • Drawing a graph of the function can really help us see what's going on.
   • Looking at limits can also provide a better understanding. For example, checking out what happens as we get close to a vertical asymptote using $\lim_{x \to a} f(x)$ can clear things up.