Calculus is the study of change. To understand change, we first need to understand how functions behave as they get 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 word here is approaches. We don't actually care what the function's value is at that number, only what it's heading towards.

Imagine walking towards a wall. With every step, you get closer. A limit is like asking: if you could keep taking smaller and smaller steps forever, where would you end up? You would end up right at the wall. The limit is the wall's position, even if you never technically touch it.

We write the limit of a function $f(x)$ as $x$ approaches a value $c$ like this:

This equation says, "The limit of $f(x)$ as $x$ approaches $c$ is $L$." It means that as the input $x$ gets arbitrarily close to $c$, the output $f(x)$ gets arbitrarily close to the value $L$.

In this graph, the function is not defined at $x=2$, which is why there's a hole. But as $x$ gets closer to 2 from either side, the function's value gets closer and closer to 4. Therefore, the limit is 4.

So how do we find these limit values? Sometimes, it's as simple as it looks.

For example, let's find the limit of $f(x) = x^2 + 5$ as $x$ approaches 3.

We just substitute 3 for $x$: $3^2 + 5 = 9 + 5 = 14$.

The limit is 14. Simple enough. But what happens when direct substitution gives us a problem?

Consider the function from our graph earlier, $f(x) = (x^2 - 4) / (x - 2)$. If we try to plug in $x=2$, we get:

This result, $0/0$, is called an indeterminate form. It doesn't mean the limit is zero or that it doesn't exist. It's a signal that we need to do more work. We need to use some algebraic techniques.

Since we are only interested in what happens as $x$ approaches 2, not at $x=2$, the $(x-2)$ terms are never zero. So we can safely cancel them.

Now we can use direct substitution on the simplified function: $2 + 2 = 4$. This confirms the limit we saw in the graph.

Let's find the limit:

Direct substitution gives $0/0$. To simplify, we multiply the numerator and denominator by the conjugate of the numerator, which is $\sqrt{x+1} + 1$.

This simplifies the numerator:

We can cancel the $x$ terms:

Now, using direct substitution:

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

We use a minus sign for the limit from the left (approaching with values less than $c$) and a plus sign for the limit from the right (approaching with values greater than $c$).

For a regular two-sided limit to exist, the left-hand limit and the right-hand limit must both exist and be equal.

$\lim_{x \to c} f(x) = L$ if and only if $\lim_{x \to c^-} f(x) = L$ and $\lim_{x \to c^+} f(x) = L$.

In the graph above, the left and right-hand limits at $x=2$ are different. Therefore, the overall limit $\lim_{x \to 2} f(x)$ does not exist.

What if a function grows without bound as it approaches a point? This is called an infinite limit. Consider the function $f(x) = 1/x^2$ as $x$ approaches 0.

As $x$ gets closer to 0 (from either side), $x^2$ becomes a very tiny positive number. Dividing 1 by a tiny positive number results in a very large positive number. The function is heading towards infinity.

It's important to note that when a limit equals $\infty$ or $-\infty$, it technically means the limit does not exist, because infinity is not a number. But we write it this way to describe the specific way it doesn't exist: the function is growing without bound.

The idea of limits leads directly to the concept of continuity. Informally, a function is continuous if you can draw its graph without lifting your pencil from the paper. There are no breaks, jumps, or holes.

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

If any of these conditions fail, the function has a discontinuity at that point.

Understanding limits and continuity is the first major step in calculus. It sets the stage for defining the rate of change (derivatives) and the accumulation of quantities (integrals).