Introduction to Limits: A First Look at Calculus for 2026

Limits are the gateway to calculus. Every derivative and every integral is defined in terms of a limit. Yet the topic intimidates most students because the notation looks unfamiliar and the idea feels abstract.

In this guide you will see that limits are simpler than they look. A limit asks one question: as x gets close to a number, what is f(x) approaching? Once that idea clicks, evaluating limits is a procedure with three or four cases.

What a Limit Is

The notation

\[\lim_{x \to a} f(x) = L\]

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means: as x approaches the value a, f(x) approaches the value L.

It does not mean f(a) = L. The function might not even be defined at a. A limit looks at what f does near a, not at a.

f(x) = (x² − 1)/(x − 1). At x = 1, the formula is 0/0 (undefined). But as x gets close to 1, the function simplifies to x + 1, which approaches 2. So the limit as x → 1 is 2, even though f(1) is undefined.

The Three Ways to Evaluate a Limit

Method 1: Plug In (When It Works)

If the function is continuous at a, just plug a into the formula.

lim (x → 3) of (2x + 5) = 2(3) + 5 = 11.

lim (x → 2) of (x² − 4x + 1) = 4 − 8 + 1 = −3.

For polynomials, exponentials, rationals (where denominator is nonzero), and most trig and log functions, plug-in works at points in the domain.

Method 2: Simplify First

When plug-in gives 0/0, factor and cancel.

lim (x → 1) of (x² − 1)/(x − 1).
Plug-in: 0/0. Factor: (x − 1)(x + 1)/(x − 1). Cancel: x + 1. Plug in x = 1: 2.

lim (x → 2) of (x² − 4)/(x − 2).
Factor: (x − 2)(x + 2)/(x − 2). Cancel: x + 2. Plug in: 4.

Method 3: Rationalize or Use Conjugates

When you have a square root and 0/0, multiply by the conjugate.

lim (x → 0) of (√(x + 4) − 2)/x.
Plug-in: 0/0.
Multiply numerator and denominator by (√(x + 4) + 2):
Numerator becomes (x + 4) − 4 = x. Denominator becomes x(√(x + 4) + 2).
Cancel x: 1/(√(x + 4) + 2).
Plug in x = 0: 1/(√4 + 2) = 1/4.

One-Sided Limits

A one-sided limit looks at x approaching from only one direction.

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• lim (x → a⁻) f(x): x approaches a from the left (x < a).
• lim (x → a⁺) f(x): x approaches a from the right (x > a).

The two-sided limit lim (x → a) f(x) exists only if the left and right limits both exist and are equal.

Step function: f(x) = 1 for x < 0, f(x) = 2 for x ≥ 0.
lim (x → 0⁻) f(x) = 1.
lim (x → 0⁺) f(x) = 2.
The two-sided limit does not exist.

Limits at Infinity

A limit at infinity asks what f does for very large (or very negative) x.

For rational functions:

• If the numerator has lower degree than the denominator, limit is 0.
• If the same degree, limit is the ratio of leading coefficients.
• If the numerator has higher degree, limit is ±∞.

lim (x → ∞) of (3x² + 1)/(x² + 5) = 3/1 = 3.
lim (x → ∞) of (x + 1)/(x² + 1) = 0.
lim (x → ∞) of (x³ + 1)/(x + 1) = ∞.

Memorize these three rules. They cover almost every infinity-limit problem in AP Calculus AB.

Indeterminate Forms

Some plug-in results require more work. They are called indeterminate forms:

• 0/0
• ∞/∞
• 0 · ∞
• ∞ − ∞
• 0⁰, 1^∞, ∞⁰

When you get one of these, do not stop. Simplify, rationalize, or use L’Hôpital’s rule (in AP Calc).

0/0 vs. non-zero/0:

• 0/0 is indeterminate. Simplify.
• 5/0 (or any non-zero over 0) is not indeterminate. It means the function has a vertical asymptote at that point; the limit is +∞ or −∞ (check the sign from each side).

Continuity

A function is continuous at a if:

1. f(a) is defined.
2. lim (x → a) f(x) exists.
3. lim (x → a) f(x) = f(a).

In plain English: no holes, no jumps, no asymptotes at a.

A function is continuous on an interval if it is continuous at every point. Polynomials are continuous everywhere. Rational functions are continuous everywhere their denominator is nonzero.

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Worked Examples

lim (x → 2) of (x² + 3x − 1).
Polynomial → plug in: 4 + 6 − 1 = 9.

lim (x → 1) of (x² − 1)/(x − 1).
Factor: x + 1. Plug in: 2.

lim (x → 0) of sin(x)/x.
A special limit; equals 1. (Memorize this one.)

lim (x → ∞) of (4x² − x + 7)/(2x² + 5).
Same degree → ratio of leading coefficients = 4/2 = 2.

lim (x → 3⁻) of 1/(x − 3).
Denominator approaches 0 from the negative side. Result: −∞.

lim (x → 3⁺) of 1/(x − 3).
Denominator approaches 0 from the positive side. Result: +∞.

A Quick Decision Tree

1. Plug in. If you get a real number, done.
2. If you get 0/0, factor or rationalize.
3. If you get non-zero/0, check sign for ±∞.
4. If x → ∞, use the rational-function rules.
5. If you have an indeterminate form and have learned L’Hôpital’s rule, apply it.

Common Mistakes

1. Stopping at 0/0. That is a starting point, not an answer.
2. Treating non-zero/0 as undefined. It usually means the limit is infinite, not nonexistent.
3. Mixing up one-sided and two-sided limits. The two-sided limit requires agreement from both sides.
4. Confusing the limit with the function value. A function can have a hole at a and still have a limit at a.
5. Forgetting to factor. Always factor when you see 0/0 with polynomials.

A Quick Cheat Sheet

Frequently Asked Questions

Are limits on the SAT?
No. Limits are a calculus topic. The SAT does not test calculus.

Are limits on AP Calculus?
Yes. Limits and continuity make up roughly 10% of AP Calculus AB and 10% of AP Calculus BC.

Do I need to know L’Hôpital’s rule from the start?
No. Most introductory limit problems can be solved by factoring or rationalizing. L’Hôpital comes after derivatives.

What if a limit gives different left and right values?
The two-sided limit does not exist. State that as your answer.

Is the limit always equal to the function value?
Only when the function is continuous at that point. Limits are about behavior near a point, not at the point.

Closing Thought

Limits are a two-word question (what’s it approaching?) and a four-step procedure (plug in, factor, rationalize, check infinity). Once you see that, the topic stops being mysterious. Drill the worked examples, memorize sin(x)/x → 1, and you have a solid foundation for the rest of calculus.

For more practice, browse our Calculus worksheets and our full Math Topics library. When you are ready for a structured workbook, our Calculus collection covers limits, derivatives, and integrals in depth.

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