Unit 1 · Topic 1.2

1.2 Defining Limits and Using Limit Notation

A limit describes the value a function's outputs get close to as the input gets close to some number. It is the language every later idea in calculus is built on. The key point is that a limit cares about values near c, not the value at c.

Key terms

limit
limit notation
x approaches c
limit value vs. function value

What the notation says

You write lim (x→c) f(x) = L and read it as “the limit of f of x as x approaches c equals L.” It means you can make f(x) as close to L as you want by taking x close enough to c, with x not equal to c.

That last part matters. When you find a limit, you never look at x = c itself. You only care about the behavior of f on both sides, very close to c.

In a real setting, read it as a sentence. If P(t) is the price of a ticket, in dollars, t days before a concert, then lim (t→5) P(t) = 40 says that as t gets close to 5, the ticket price gets close to 40 dollars. It doesn't say what the price is exactly 5 days before.

Limit value vs. function value

Because the limit ignores x = c, three different situations can happen:

f(c) exists and equals the limit. This is the “nice” case, like lim (x→2) x² = 4 = f(2).
f(c) does not exist, but the limit does. For f(x) = (x² − 9) / (x − 3), f(3) is undefined (0/0), yet the outputs near 3 approach 6.
f(c) exists but is a different number. If g(x) = x + 1 for x ≠ 2 and g(2) = 7, then lim (x→2) g(x) = 3 even though g(2) = 7.

One-sided limits

Sometimes you only look at one side. lim (x→c⁻) f(x) is the left-hand limit, using x-values less than c. lim (x→c⁺) f(x) is the right-hand limit, using x-values greater than c.

The two-sided limit exists exactly when both one-sided limits exist and are equal. If the left side heads to 2 and the right side heads to 5, the limit does not exist (often written DNE).

When the outputs blow up

If f(x) grows without bound near c, you may write lim (x→c) f(x) = ∞. This is a description of how the limit fails, not a real number. Strictly, the limit still does not exist, but the ∞ tells you why. Topic 1.14 covers this case.

What you won't be tested on

Some textbooks give a formal epsilon-delta definition of a limit. The AP exam does not test it. You need the idea (close enough to c forces close to L), the notation and the ability to find limits.

Worked examples

Try each one yourself first, then open the solution.

Example 1
A limit where the function is undefined
Let f(x) = (x² − 9) / (x − 3). Find f(3) and lim (x→3) f(x).
Show the solutionHide the solution
1. Step 1: Plugging in x = 3 gives (9 − 9) / (3 − 3) = 0/0, so f(3) is undefined.
2. Step 2: For x ≠ 3, factor: (x − 3)(x + 3) / (x − 3) = x + 3. Canceling is allowed because x is never equal to 3 in a limit.
3. Step 3: As x approaches 3, x + 3 approaches 6.
Answer: f(3) does not exist, but lim (x→3) f(x) = 6.
Example 2
Trap: the function value is not the limit
g(x) = x + 1 when x ≠ 2, and g(2) = 7. Find lim (x→2⁻) g(x), lim (x→2⁺) g(x), lim (x→2) g(x) and g(2).
Show the solutionHide the solution
1. Step 1: Near 2 but not at 2, g(x) = x + 1. From the left, values like 1.9, 1.99 give outputs 2.9, 2.99, heading to 3.
2. Step 2: From the right, 2.1, 2.01 give 3.1, 3.01, also heading to 3.
3. Step 3: Both one-sided limits equal 3, so the two-sided limit is 3.
4. Step 4: The function value is whatever the rule says at x = 2 exactly: g(2) = 7.
Answer: Both one-sided limits are 3, so lim (x→2) g(x) = 3, while g(2) = 7.

Common mistakes

Saying the limit doesn't exist because f(c) is undefined. A hole at x = c says nothing about whether the limit exists.
Writing lim (x→c) f(x) = f(c) automatically. That only works when f is continuous at c.
Forgetting to check both sides. A two-sided limit needs the left-hand and right-hand limits to match.

On the exam

Multiple-choice questions often show a graph and ask which statement is true, mixing up f(c) and lim (x→c) f(x). Read each one carefully and check the dot (the actual value) separately from where the curve heads.
Write limits with full notation, including lim and the arrow. An answer like “= 6” without the limit statement can lose credit on free response.

Connected topics

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