Can a limit exist if the function is undefined at that point?

Yes! That is the main power of limits. The limit only cares about what the function is approaching, not what it actually is at that exact spot.

What does it mean to approach from the left or right?

Approaching from the left means looking at $x$ values slightly smaller than your target (like $1.9, 1.99, 1.999$). Approaching from the right means looking at values slightly larger (like $2.1, 2.01, 2.001$). For a standard limit to exist, both sides must point to the exact same destination.

What if the left and right sides approach different numbers?

If the two sides do not agree on the destination, we say the limit "does not exist" (often abbreviated as DNE).

What is a limit in plain language?

In plain language, a limit is the value that a mathematical function attempts to reach as you plug in numbers closer and closer to a specific point. It is not about what happens exactly at that target point, but rather about the journey getting there.

Think of it like walking toward a wall. If you step halfway to the wall, then halfway again, and keep doing this, your destination (the limit) is the wall itself. Even if technically your steps get infinitely small before you ever touch the wall, everyone watching knows exactly where you are headed. In calculus, limits help us describe this intended destination.

What is a limit mathematically?

In calculus, we write a limit like this: $\lim_{x \to a} f(x) = L$ . This is read as "the limit of $f(x)$ as $x$ approaches $a$ is $L$ ." It means that as your input variable $x$ gets extremely close to the number $a$ (from both sides), your output value $f(x)$ gets extremely close to the number $L$ .

Why do limits work and why do we need them?

Limits work because they bypass the exact point in question. We need them because algebra sometimes breaks down. For example, if you try to calculate speed at exactly one frozen instant in time, you get zero distance divided by zero time, which is undefined. Limits let us see what the speed should be at that instant by looking at infinitesimally small time intervals around it.

How to recognize a limit problem

You will recognize limit problems by the notation $\lim$ . Sometimes, you can just plug the target number directly into the function to find the answer. Other times, plugging in the number gives you $0/0$ , which mathematicians call an "indeterminate form." That is your cue to use algebra to simplify the expression before trying again.

Where students slip up

The most common mistake is assuming that the limit is always the same as the function's value at that point. Students often confuse $f(a)$ with the limit as $x \to a$ . Remember, a function can have a "hole" (be undefined) at $x = 3$ , but the limit as $x$ approaches $3$ can still exist perfectly well.

Worked through

Evaluate the limit: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

First, try plugging in $x = 2$ directly. You get $\frac{2^2 - 4}{2 - 2} = \frac{0}{0}$ . This is an indeterminate form, which means we have more work to do.

Next, try factoring the numerator. The expression $x^2 - 4$ is a difference of squares: $x^2 - 4 = (x - 2)(x + 2)$

Now rewrite the original limit with the factored form: $\lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2}$

Notice that we have an $(x - 2)$ on the top and bottom. Because we are taking a limit as $x$ approaches $2$ (meaning $x$ is not exactly $2$ ), we are not dividing by zero when we cancel them out: $\lim_{x \to 2} (x + 2)$

Finally, plug in $x = 2$ into our simplified function: $2 + 2 = 4$

The limit is $4$ .

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Where this comes from: OpenStax Calculus Volume 1, Chapter 2: Limits · Khan Academy: AP Calculus AB, Unit 1: Limits and Continuity