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

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

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.

• 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.

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

How do you introduce limits? A good way to introduce limits is with a table of values and a graph before any formal notation. Pick f(x) = (x² − 1)/(x − 1), try x = 0.9, 0.99, 0.999 and 1.1, 1.01, 1.001, and watch the outputs approach 2, even though f(1) is undefined. This shows that a limit describes what f(x) approaches near a point, not its value at the point.

What is a limit in calculus? In calculus, a limit is the value a function approaches as the input gets closer and closer to a certain number. The notation lim (x → a) f(x) = L means f(x) gets arbitrarily close to L as x approaches a. For example, lim (x → 3) of (2x + 5) = 11. Limits are the foundation for both derivatives and integrals.

How are limits and continuity related? A function is continuous at x = a when three things hold: f(a) is defined, the limit as x → a exists, and the limit equals f(a). So limits are how continuity is defined. For example, (x² − 4)/(x − 2) has a limit of 4 as x → 2, but it is not continuous there because the function is undefined at x = 2.

How are limits used to define derivatives? A derivative is defined as a limit of slopes: f′(x) = lim (h → 0) [f(x + h) − f(x)] ÷ h. It takes the slope between two nearby points and lets the gap shrink to zero. For f(x) = x², the expression simplifies to 2x + h, which approaches 2x as h → 0, so the derivative of x² is 2x.

What should you learn first in basic calculus limits? In basic calculus, start with the three main ways to evaluate a limit: plug in the value when the function is continuous, factor and cancel when you get 0/0, and multiply by the conjugate when a square root causes 0/0. Then learn one-sided limits and limits at infinity. For example, lim (x → 0) of (√(x + 4) − 2)/x equals 1/4 after rationalizing.

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 (resource currently unavailable) and our full Math Topics library. When you are ready for a structured workbook, our Calculus collection covers limits, derivatives, and integrals in depth.