Calculus is the study of change, and to understand change, we first need to understand the idea of getting infinitely close to something. This is the core idea of a limit. A limit describes the value a function approaches as its input gets closer and closer to a specific number.

Notice the word “approaches.” The function doesn’t actually have to reach that value. The limit is all about the journey, not the destination. It's the value we'd expect the function to have if we followed its path right up to the edge of a certain point.

We can find limits in a couple of ways. The most intuitive way is by looking at a graph. Let's see what that looks like.

In the graph above, the function is undefined right at $x=1$. There’s a hole in the graph. But we can still talk about the limit. As we trace the line from the left side, getting closer and closer to $x=1$, what y-value are we approaching? We're heading towards $y=2$. If we trace the line from the right side towards $x=1$, we are also heading towards $y=2$. Since the function approaches 2 from both sides, we say the limit of the function as $x$ approaches 1 is 2. We write this as:

Another way to see this is by plugging in numbers that are extremely close to our target value. This is the numerical approach.

As you can see from the table, as $x$ gets closer to 1 from either below (0.999) or above (1.001), the value of $f(x)$ gets closer and closer to 2. The numerical and graphical methods both point to the same limit.

What happens when a function approaches different values from the left and the right? In these cases, we use one-sided limits. A left-hand limit is the value a function approaches as $x$ comes from the negative side (smaller numbers). A right-hand limit is for when $x$ comes from the positive side (larger numbers). We use a little minus or plus sign to show which side we're talking about.

For a regular limit to exist, the left-hand limit and the right-hand limit must be equal. If they aren't, the overall limit does not exist at that point.

In this graph, as $x$ approaches 1 from the left, the function heads toward $y=1$. But as $x$ approaches 1 from the right, the function heads toward $y=2$. Because the left and right-hand limits are different, the overall limit as $x$ approaches 1 does not exist.

Sometimes, a function doesn't approach a specific number but instead grows larger or smaller without any boundary. We call these infinite limits. For example, the function $f(x) = 1/x^2$ grows infinitely large as $x$ gets close to 0. We'd say the limit is infinity.

The idea of limits leads directly to the concept of continuity. A function is continuous if you can draw its graph without lifting your pencil from the paper. There are no jumps, holes, or gaps. For a function to be continuous at a point $c$, three conditions must be met:

If any of these fail, the function is discontinuous at that point. The first graph we saw had a hole, which is a removable discontinuity. The second graph had a jump, which is a non-removable discontinuity.

Continuous functions have a very useful property, described by the Intermediate Value Theorem. It says that for any continuous function on an interval, the function must take on every single value between its starting and ending points. For instance, if you start a hike at an elevation of 100 feet and end at 500 feet, and your path is continuous (you don't teleport!), you must have passed through every elevation in between, like 250 feet and 412.3 feet.

This theorem is more powerful than it looks. It guarantees that continuous functions don't skip values, which is a cornerstone for many proofs in calculus.

Let's test your understanding of these foundational concepts.

Understanding limits and continuity is the first major step into the world of calculus. It allows us to analyze the behavior of functions with precision and sets the stage for exploring rates of change and accumulation.