Calculus is the study of change, and at its heart is a simple idea: getting incredibly close to something to understand its behavior. This idea is called a limit.

A limit tells us what value a function gets closer and closer to as its input approaches a specific number. The key here is approaching. We don't care what happens exactly at the number, only what happens in its immediate neighborhood.

Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$. If you try to plug in $x=2$, you get $\frac{0}{0}$, which is undefined. The function has a hole at that exact spot.

But what happens as $x$ gets close to 2? If you plug in 1.9, you get 3.9. If you plug in 2.1, you get 4.1. The closer you get to 2, the closer the function's value gets to 4. We say the limit of $f(x)$ as $x$ approaches 2 is 4. In mathematical notation, we write this as:

This reads: "The limit as $x$ approaches 2 of $f(x)$ equals 4."

Sometimes, finding a limit is easy. For many functions, you can simply substitute the value into the function.

For example, to find the limit of $f(x) = x^2 + 3$ as $x$ approaches 5:

This is called direct substitution, and it works for any polynomial and many other simple functions. But as we saw earlier, it doesn't always work. When direct substitution gives you an undefined form like $\frac{0}{0}$, you need other techniques.

One common method is factoring. We can simplify the function we looked at before:

As long as $x$ is not equal to 2, we can cancel the $(x-2)$ terms, leaving us with $f(x) = x + 2$. Since limits are about approaching a value, not being exactly at it, we can use this simplified function to find the limit:

This confirms our earlier result.

Sometimes a function behaves differently depending on which direction you approach a point from. This is where one-sided limits come in.

• A left-hand limit approaches a value from the negative side (numbers smaller than the target).
• A right-hand limit approaches from the positive side (numbers larger than the target).

We use a superscript minus for the left and a plus for the right.

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

We can also ask what happens to a function as its input grows infinitely large, either positive or negative. This is called a limit at infinity.

Consider the function $f(x) = \frac{1}{x}$. As $x$ gets very large, like 1,000 or 1,000,000, the value of $f(x)$ gets closer and closer to 0. The same thing happens as $x$ becomes a very large negative number.

Limits also help us define a very important property of functions: continuity. A function is continuous at a point if you can draw it without lifting your pen from the paper. Formally, a function $f(x)$ is continuous at a point $c$ if three conditions are met:

1. $f(c)$ is defined (the point exists).
2. $\lim_{x \to c} f(x)$ exists (the function approaches a single value).
3. $\lim_{x \to c} f(x) = f(c)$ (the limit equals the actual value).

If any of these fail, the function has a discontinuity. The function we examined earlier, $f(x) = \frac{x^2-4}{x-2}$, has a removable discontinuity (a hole) at $x=2$ because the first condition fails. Other types include jump discontinuities, where the left and right limits are different, and infinite discontinuities, where the function goes to infinity.

Understanding these ideas sets the stage for the rest of calculus.