In calculus, a limit is all about getting incredibly close to a point without necessarily landing right on it. Think of it as walking toward a wall. You can get closer and closer, taking smaller and smaller steps, but you never actually touch the wall. A limit describes the location you're approaching.

Mathematically, a limit tells us the value a function approaches as its input gets closer to a certain number. We write it like this:

This is read as "the limit of $f(x)$ as $x$ approaches $c$ equals $L$." It means that as the value of $x$ gets arbitrarily close to $c$, the value of $f(x)$ gets arbitrarily close to $L$.

For many simple functions, this is straightforward. Consider the function $f(x) = x + 2$. What is the limit as $x$ approaches 3? As $x$ gets closer to 3, $f(x)$ gets closer to 5. So, we say the limit is 5.

But what about a function like this?

If we try to find the limit as $x$ approaches 2, we run into a problem. Plugging in $x=2$ gives us $\frac{0}{0}$, which is undefined. The function doesn't exist at $x=2$. However, the limit might. Limits are concerned with the neighborhood around a point, not the point itself.

So, how do we find the limit of that tricky function? One common technique is to simplify the expression algebraically.

We can factor the numerator:

Since we are only interested in what happens as $x$ approaches 2, and not at $x=2$, we know that $x-2$ is not zero. This means we can safely cancel the $(x-2)$ terms.

Now, it's easy to see that as $x$ gets closer to 2, the function's value gets closer to $2+2=4$. So, the limit is 4.

This highlights a key idea: the limit of a function at a point can exist even if the function itself is undefined at that point.

Sometimes a function behaves differently depending on which direction you approach a point from. This leads to the idea of one-sided limits.

• A right-hand limit ($x \to c^+$) means you're only considering values of $x$ that are greater than $c$.
• A left-hand limit ($x \to c^-$) means you're only considering values of $x$ that are less than $c$.

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

Limits can also describe a function's behavior as it grows without bound or as its input gets infinitely large.

An infinite limit occurs when the function's value increases or decreases forever as $x$ approaches a certain number. This often happens at a vertical asymptote. For example, consider the function $f(x) = \frac{1}{x^2}$. As $x$ gets closer and closer to 0 from either side, the value of $f(x)$ gets huge.

A limit at infinity describes the end behavior of a function—what value it approaches as $x$ goes to positive or negative infinity. This value corresponds to a horizontal asymptote. For example, let's look at $f(x) = \frac{1}{x}$. As $x$ gets extremely large, the value of $\frac{1}{x}$ gets extremely small, approaching 0.

The concept of continuity is closely tied to limits. Intuitively, a function is continuous if you can draw its graph without lifting your pen from the paper. There are no holes, jumps, or vertical asymptotes.

More formally, a function $f(x)$ is continuous at a point $x=c$ if it meets three conditions:

If any of these conditions fail, the function is discontinuous at that point. There are a few common types of discontinuities.

This is the type of discontinuity we saw earlier. We can "remove" it by defining $f(2)=4$.

Understanding limits and continuity is the first major step into calculus. These ideas form the bedrock upon which derivatives are built, allowing us to precisely define rates of change.