Understanding limits is really important when you're learning about continuity in functions, especially in calculus classes at the AS-Level.

What Are Limits?

Simply put, limits help us find out what a function is doing as it gets closer to a certain point, even if it doesn’t actually reach that point. Knowing this is key to figuring out if a function is continuous at a specific spot on its graph.

The Connection Between Limits and Continuity

1. What is Continuity?
   A function, which we call $f(x)$, is continuous at a point $c$ if three things happen:

   • First, $f(c)$ has to exist (that means it’s defined).
   • Second, the limit of $f(x)$ as $x$ gets closer to $c$ has to exist.
   • Third, this limit has to be the same as the value of the function: $\lim_{x \to c} f(x) = f(c)$.

   If any of these rules don’t work, the function is not continuous at that point. We often think about this in our lessons.

2. Looking at One-Sided Limits
   Sometimes, it helps to check one side at a time—like the left side ($\lim_{x \to c^-} f(x)$) and the right side ($\lim_{x \to c^+} f(x)$). If both sides match and equal $f(c)$, we can say that the function is continuous at that point. This becomes handy when a function looks different on each side of a specific point.

Why Are Limits Important?

• Handling Undefined Points:
  There are times when functions can’t give a value at certain points, like when you have a division by zero. Limits help us figure out what value the function is getting close to as we approach that tricky point. For example, in the function $f(x) = \frac{x^2 - 1}{x - 1}$, $f(1)$ is undefined, but as $x$ gets closer to $1$, $\lim_{x \to 1} f(x)$ gets close to $2$. This means there's a removable break in the function at $x=1$.

• Looking at Behavior Near Infinity:
  Limits also help us understand what happens as we get really big values (infinity). Knowing the limits as $x$ approaches infinity can help us sketch and understand the overall shape of the graph.

How We Use Limits

In Year 12 classes, we often use limits to graph both continuous and discontinuous functions. It’s amazing how a single limit can change our view of how a function works. For instance, learning about limits can lead us into deeper talks about real-life things like speed or population growth, all of which use the basic idea of limits to help us understand continuity.

In short, limits are essential for understanding continuity in functions. They help us see how functions act around certain points, showing us where they might break or stay smooth. It’s a powerful tool in calculus that really boosts our math skills!