When you start studying rational functions in Year 12 Maths, it’s important to get a good grasp on asymptotes. These are special lines that help you see how these functions behave, especially when they get close to certain numbers or keep going to infinity. Let's break down how using limits can help you understand these asymptotes better.

Vertical Asymptotes

Vertical asymptotes happen when the function goes up to infinity or down to negative infinity as the input gets close to a certain value. To find them, you usually look at the bottom part (denominator) of a rational function. For example, take a look at this function:

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

1. Find Where the Denominator is Zero: To do this, set the denominator equal to zero. For our example, solve $x - 2 = 0$. You get $x = 2$.

2. Check Limits: Now, look at what happens as $x$ gets close to the asymptote:

   • When $x$ approaches 2 from the left: $\lim_{x \to 2^-} f(x) = -\infty$
   • When $x$ approaches 2 from the right: $\lim_{x \to 2^+} f(x) = \infty$

This tells you that there is a vertical asymptote at $x = 2$, where the function goes up to infinity in one direction and down in the other.

Horizontal Asymptotes

Horizontal asymptotes show what happens when $x$ goes toward infinity or negative infinity. Here, you mainly look at the powers (or degrees) of the polynomials in the top (numerator) and bottom (denominator). Using limits can help you with this:

1. Compare Degrees: For a function like $f(x) = \frac{2x^2 + 3}{x^2 - 5}$ you look at the degrees of the top and bottom.

   • Both are degree 2, so you find the horizontal asymptote by dividing the leading numbers (coefficients):
   • $y = \frac{2}{1} = 2$

2. Check with Limits: Confirm this using a limit:

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

So, the horizontal asymptote is $y = 2$.

Behavior at Infinity

Understanding what limits mean at infinity helps you not only with horizontal asymptotes but also gives you an idea of how the function behaves over time. As you figure out these details, it gives you a clearer picture of what the graph will look like overall.

In short, limits are a powerful tool to help you analyze both vertical and horizontal asymptotes in rational functions. The more you practice, the easier these concepts will become, making it simpler to graph and understand functions!