Calculus is the study of change, and to understand change, we first need to talk about getting incredibly close to a specific point. This is the idea of a limit. A limit tells us what value a function approaches as its input gets closer and closer to a certain number.

The key idea is that we don't care what happens at the exact point, only what happens in the neighborhood surrounding it. We write the limit of a function $f(x)$ as $x$ approaches a number $c$ like this:

This means that as $x$ gets closer and closer to $c$ (from both sides), the value of $f(x)$ gets closer and closer to $L$.

So how do we calculate $L$? Sometimes, it's as simple as plugging the number in. This is called direct substitution.

For example, let's find the limit of $f(x) = x^2 + 1$ as $x$ approaches 3.

Easy enough. But what about the function from our graph earlier, $f(x) = \frac{x^2 - 4}{x - 2}$? If we try to plug in $x=2$, we get:

This result, $\frac{0}{0}$, is called an indeterminate form. It's a sign that we need to do more work. It doesn't mean the limit doesn't exist. In this case, we can simplify the function by factoring.

Since we only care about what happens near $x=2$, not at $x=2$, we know that $x-2$ is not zero. That means we can safely cancel the $(x-2)$ terms.

This confirms what we saw on the graph. Even though the function is undefined at $x=2$, its limit is 4.

Continuity is a straightforward idea. A function is continuous if you can draw its graph without lifting your pencil from the paper. There are no holes, jumps, or gaps.

This intuitive idea has a formal definition that ties directly back to limits. A function $f(x)$ is continuous at a point $c$ if three conditions are met:

1. The function is defined at $c$. In other words, $f(c)$ exists.
2. The limit of the function exists at $c$. So, $\lim_{x \to c} f(x)$ exists.
3. The limit at $c$ is equal to the function's value at $c$.

Think back to our two examples. The function $f(x) = x^2 + 1$ is continuous everywhere. At $x=3$, the limit was 10, and the function's value $f(3)$ is also 10.

However, the function $f(x) = \frac{x^2 - 4}{x - 2}$ is not continuous at $x=2$. It has a limit there (which is 4), but the function itself is undefined. It fails the first condition.

Understanding limits and continuity is the first major step in calculus. These concepts allow us to handle the idea of 'instantaneous' change, which is what we will explore when we get to derivatives.