In calculus, we're often interested in how things change. But to understand change, we first need to understand the idea of getting "infinitely close" to a point without actually being there. This is the core concept of a limit.

A limit tells us the value a function approaches as its input gets closer and closer to a certain number. It's less about the destination and more about the journey's direction. We don't care what the function's value is at the point, only what it's heading towards from either side.

The notation for a limit looks like this:

Imagine a graph of a function with a tiny hole at $x = 2$. The function is defined everywhere else, but at that exact spot, it's undefined. A limit allows us to talk about the height the graph is approaching as we get closer to that hole from both the left and the right.

So how do we find these values? There are a few key techniques.

Direct Substitution For many simple functions, finding the limit is as easy as plugging the number into the function. If the function is continuous (meaning it has no breaks, jumps, or holes), the limit at a point is simply the function's value there.

Factoring and Canceling Sometimes, direct substitution gives us an "indeterminate form" like $\frac{0}{0}$. This doesn't mean the limit doesn't exist. It's a sign we need to do more work, usually by simplifying the expression first.

Other algebraic techniques, like rationalizing an expression with a square root, can also be used to simplify the function before substituting.

Sometimes a function approaches different values from the left side versus the right side of a point. This is where one-sided limits are useful.

A left-hand limit is written as $\lim_{x \to c^-} f(x)$ and considers only values of $x$ that are less than $c$. A right-hand limit is written as $\lim_{x \to c^+} f(x)$ and considers only values of $x$ that are greater than $c$.

For a general limit to exist, the left- and right-hand limits must be equal. If they're different, the overall limit at that point does not exist.

Limits can also describe behavior that isn't a single finite number. We have two cases: infinite limits and limits at infinity.

Infinite Limits An infinite limit occurs when the function's values grow without bound as $x$ approaches a certain number. This often happens at a vertical asymptote.

Limits at Infinity This type of limit describes the end behavior of a function. What happens to the output as the input $x$ becomes extremely large (either positive or negative)?

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

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

If any of these fail, the function has a discontinuity at that point. The types of discontinuities we've seen—holes and jumps—are defined by how these limit conditions fail. Understanding limits is the key to precisely describing where a function is connected and where it is broken.

Now, let's test your understanding of these foundational concepts.