# Limit Laws: Sum, Product, Quotient and Power Rules for Limits

> The limit laws let you split a limit into smaller limits. Every law with its conditions, worked examples and the mistakes that break them.

Source: https://practicecalculus.com/rules/limit-laws
Updated: 2026-11-03

$$ \lim_{x \to a}\,[f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) $$

## At a glance

- **Applies to:** Limits of sums, differences, multiples, products, quotients, powers and roots
- **Key condition:** Each individual limit must exist as a finite number
- **Quotient law needs:** The limit of the denominator to be nonzero
- **In words:** The limit of a combination is the same combination of the limits
- **Consequence:** Polynomials and rational functions can be evaluated by direct substitution

## Key takeaways

- If $\lim f$ and $\lim g$ both exist, you can take the limit of each piece and combine the answers.
- The quotient law only works when the limit of the denominator is not $0$.
- When a law does not apply, such as a $\frac{0}{0}$ quotient, simplify the expression first.
- The limit laws are the reason direct substitution works for polynomials.

The **limit laws** are the algebra rules for limits. They say that if two functions each approach a finite limit, then their sum, difference, product, quotient and powers approach the matching combination of those limits. They turn one hard limit into several easy ones.

## The limit laws

Suppose $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, where $L$ and $M$ are real numbers, and $c$ is a constant.

| Law | Statement |
| --- | --- |
| Constant | $\lim_{x \to a} c = c$ |
| Identity | $\lim_{x \to a} x = a$ |
| Sum | $\lim_{x \to a} [f(x) + g(x)] = L + M$ |
| Difference | $\lim_{x \to a} [f(x) - g(x)] = L - M$ |
| Constant multiple | $\lim_{x \to a} c\,f(x) = cL$ |
| Product | $\lim_{x \to a} f(x)\,g(x) = LM$ |
| Quotient | $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \ne 0$ |
| Power | $\lim_{x \to a} [f(x)]^n = L^n$ for a positive integer $n$ |
| Root | $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L}$, with $L > 0$ if $n$ is even |

A related rule handles compositions: if $\lim_{x \to a} g(x) = M$ and $f$ is continuous at $M$, then $\lim_{x \to a} f(g(x)) = f(M)$. This is why you can move a limit inside $\sin$, $e^x$, $\ln$ or a square root, as long as the inner limit lands where the outer function is [continuous](/blog/continuity).

## Worked examples

**Example 1: a polynomial.** Apply the sum, constant multiple and power laws one term at a time:

$$\lim_{x \to 2} (3x^2 - 5x + 1) = 3(2)^2 - 5(2) + 1 = 12 - 10 + 1 = 3$$

**Example 2: a quotient.** The denominator approaches $1 + 1 = 2 \ne 0$, so the quotient law applies:

$$\lim_{x \to 1} \frac{x^2 + 3}{x + 1} = \frac{1 + 3}{1 + 1} = 2$$

**Example 3: combining given limits.** Suppose $\lim_{x \to 4} f(x) = 3$ and $\lim_{x \to 4} g(x) = -2$. Then

$$\lim_{x \to 4} \left[2f(x) - g(x)^2\right] = 2(3) - (-2)^2 = 2$$

$$\lim_{x \to 4} \frac{f(x)\,g(x)}{f(x) + 1} = \frac{(3)(-2)}{3 + 1} = -\frac{3}{2}$$

**Example 4: when the quotient law fails.** For $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$ the denominator approaches $0$, so the quotient law says nothing. Simplify first, then use the laws on the new expression:

$$\lim_{x \to 3} \frac{(x-3)(x+3)}{x - 3} = \lim_{x \to 3} (x + 3) = 6$$

Canceling is allowed because the limit never uses the value at $x = 3$ itself.

## Why direct substitution follows from the laws

Every polynomial is built from constants and $x$ using sums, products and powers. Applying the laws step by step always ends with the polynomial evaluated at $a$. So for any polynomial $p$:

$$\lim_{x \to a} p(x) = p(a)$$

The same is true for a rational function $\frac{p(x)}{q(x)}$ whenever $q(a) \ne 0$. That shortcut is called [direct substitution](/rules/direct-substitution), and it is the first thing to try on any limit.

## When the limit laws do not apply

The laws need every individual limit to **exist and be finite**. Watch for these situations:

- **A denominator that approaches $0$.** If the numerator also approaches $0$, you have the indeterminate form $\frac{0}{0}$. Factor, rationalize or use [L'Hôpital's rule](/rules/lhopitals-rule). If the numerator approaches a nonzero number, the limit is infinite or does not exist.
- **An infinite limit.** Forms such as $\infty - \infty$, $0 \cdot \infty$ and $\frac{\infty}{\infty}$ have no automatic value. See the guide to [limits at infinity](/blog/limits-at-infinity).
- **A limit that does not exist.** If $\lim f(x)$ does not exist, you cannot split it off. For instance, $\lim_{x \to 0} \left[\sin\frac{1}{x} - \sin\frac{1}{x}\right] = 0$, even though $\lim_{x \to 0} \sin\frac{1}{x}$ does not exist.

## Common mistakes

1. **Using the quotient law on $\frac{0}{0}$.** Writing $\frac{0}{0}$ and stopping, or calling the answer $0$ or $1$, is wrong. It signals that more algebra is needed.
2. **Splitting before checking existence.** The sum law requires both limits to exist. Check that first.
3. **Forgetting the root condition.** $\lim \sqrt{f(x)}$ needs $f(x)$ to approach a positive number (or approach $0$ from above) for the square root to make sense.
4. **Pulling a limit inside a discontinuous function.** The composition rule requires the outer function to be continuous at the inner limit.

For a full walkthrough of factoring, rationalizing and other techniques, read the guide on [how to evaluate limits](/blog/how-to-evaluate-limits). If limits are new to you, start with [what a limit is](/blog/what-is-a-limit).

## Frequently asked questions

### What are the limit laws in calculus?

They are rules that say the limit of a sum, difference, constant multiple, product, quotient, power or root equals the same combination of the individual limits, provided those limits exist.

### When can you not use the quotient law for limits?

When the denominator approaches $0$. Then the law does not apply and you must factor, rationalize, or use another method such as L'Hôpital's rule.

### Do the limit laws work if a limit is infinite?

Not directly. The laws are stated for finite limits. Expressions such as $\infty - \infty$ or $\frac{\infty}{\infty}$ are indeterminate and need extra work.

### Do limit laws work for one-sided limits?

Yes. Every limit law holds for left-hand and right-hand limits and for limits as $x \to \pm\infty$, as long as the individual limits exist.