In calculus, a limit is all about getting closer and closer to something without necessarily reaching it. Think about walking towards a wall. You can take steps that are half the remaining distance. First, you cover half the distance. Then you cover half of what's left. Then half of that, and so on. You'll get incredibly close to the wall, but you'll never technically touch it. A limit describes the value you're approaching—in this case, the wall.

Let's look at the function $f(x) = x + 2$. What value does $f(x)$ approach as $x$ gets closer to 3? It's easy to see that as $x$ approaches 3, $f(x)$ approaches 5. We write this using limit notation:

This reads as "the limit of $(x+2)$ as $x$ approaches 3 is 5." For many simple functions, finding the limit is as easy as plugging the number in. But what happens when we can't just plug it in?

Consider this function:

What is the limit as $x$ approaches 1? If we try to plug in $x=1$, we get $\frac{1-1}{1-1} = \frac{0}{0}$. This is called an indeterminate form. It doesn't mean the limit doesn't exist. It just means we have to do more work.

The function is undefined at $x=1$, but a limit doesn't care about the value at the point. It only cares about what the function is doing near the point. Let's look at a table of values close to $x=1$:

As $x$ gets closer to 1 from both sides, $f(x)$ gets closer to 2. So, the limit is 2. We can prove this algebraically by factoring the numerator:

As long as $x \neq 1$, we can cancel the $(x-1)$ terms. This leaves us with $f(x) = x+1$. Now it's easy to see that as $x$ approaches 1, the function approaches $1+1=2$. This function has a "hole" at $x=1$, but the limit still exists there.

Sometimes, a function approaches different values from the left and from the right. This brings us to the idea of one-sided limits.

For a regular limit to exist, the left-hand limit and the right-hand limit must be equal. If they are different, the overall limit does not exist.

In the graph above, as $x$ approaches 2 from the left, the function approaches 2. As $x$ approaches 2 from the right, the function approaches 4. Since $2 \neq 4$, the general limit $\lim_{x \to 2} f(x)$ does not exist.

Limits lead directly to the idea of continuity. Informally, a function is continuous if you can draw its graph without lifting your pencil from the paper. The jump in the graph above is a discontinuity. The hole in the earlier graph is also a discontinuity.

More formally, a function $f(x)$ is continuous at a point $c$ if three conditions are met:

If any of these fail, the function is not continuous at that point. Polynomials, for example, are continuous everywhere. Rational functions (fractions of polynomials) are continuous wherever their denominator is not zero.

Let's test your understanding of these foundational concepts.

Mastering limits is the first major step in understanding calculus. They provide the precise language needed to talk about change and approximation, which are the core ideas of the entire subject.