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📐 What is a Limit?

$\lim_{x \to a} f(x) = L$ — the foundation of Calculus

Calculus is often called the study of "change" — and it all begins with the concept of a limit. Derivatives, integrals, continuity... every one of these ideas is built on top of limits. So here's the most fundamental question you'll face at the start of AP Calculus: what exactly is a limit?

📌 The Basic Idea

The limit of a function at a point is the value that $f(x)$ gets closer and closer to as $x$ gets closer and closer to that point — from both sides. In other words, the limit is "where the function is heading," whether or not the function actually reaches that value at that exact point.

$$ \lim_{x \to a} f(x) = L $$
"the limit of f(x) as x approaches a is L"
💡 The Key Point
A limit describes the behavior of $f(x)$ near $x = a$ — not necessarily what happens at $x = a$. The function doesn't even need to be defined at $x = a$ for the limit to exist!

🚗 A Real-World Analogy

Imagine driving toward a city. Road signs read "City Center: 2 km... 1 km... 0.5 km..." — the numbers keep shrinking. Even if you never actually arrive (say you take an exit before reaching downtown), you can clearly tell where you were heading. That's exactly what a limit captures: how close a quantity gets to a value, regardless of whether it ever actually lands there.

📋 Estimating a Limit from a Table

One way to estimate a limit is to plug in values of $x$ that get extremely close to $a$ from both the left and the right, and see what $f(x)$ approaches.

Example 1Numerical Estimation

Estimate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ using a table of values.

$x$$f(x)$$x$$f(x)$
$1.9$$3.9$$2.1$$4.1$
$1.99$$3.99$$2.01$$4.01$
$1.999$$3.999$$2.001$$4.001$

As $x$ approaches $2$ from either side, $f(x)$ approaches $4$. Note that $f(x)$ is actually undefined at $x=2$ (the denominator becomes zero) — yet the limit still exists:

$$\lim_{x \to 2} \frac{x^2-4}{x-2} = 4$$

📊 Estimating a Limit from a Graph

Use the interactive graph below to see what happens to a curve as $x$ approaches a particular point. Try all three — pay special attention to the one with a "hole": the function isn't defined at that point, but the limit is still perfectly clear.

↔️ One-Sided Limits

For a limit to exist, $f(x)$ must approach the same value whether $x$ approaches $a$ from the left or from the right.

$$ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \quad \Longrightarrow \quad \lim_{x \to a} f(x) = L $$
⚠️ When a Limit Does Not Exist (DNE)
If the left-hand and right-hand limits are different, if the function shoots off to infinity, or if it oscillates without settling down (like $\sin(1/x)$ near $0$), then $\lim_{x \to a} f(x)$ does not exist — we write this as DNE.

🧮 How to Evaluate Limits (Quick Overview)

In AP Calculus, you'll typically try these strategies in order:

  1. Direct substitution: Plug $x = a$ directly into the function. If the result is defined, that's your answer.
  2. Factor and cancel: If you get an indeterminate form like $\frac{0}{0}$, factor the expression and cancel common terms.
  3. Rationalize: For expressions with square roots, multiply by the conjugate.
  4. Tables or graphs: If no algebraic approach works, estimate numerically or graphically.
Example 2Direct Substitution

Find $\lim_{x \to 4} (3x - 5)$.

The function is defined and continuous at $x=4$, so we can substitute directly:

$$3(4) - 5 = 12 - 5 = 7$$
Example 3Factor and Cancel

Find $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Direct substitution gives $\frac{0}{0}$ — an indeterminate form. So we factor and cancel:

$$\frac{x^2-9}{x-3} = \frac{(x-3)(x+3)}{x-3} = x+3 \quad (x \neq 3)$$

Now we can substitute directly:

$$\lim_{x \to 3}(x+3) = 6$$
🔍 Key Takeaways
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