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

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.

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.

Worked examples

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

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

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

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:

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:

The same is true for a rational function \frac{p(x)}{q(x)} whenever q(a) \ne 0. That shortcut is called 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. 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.
• 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. If limits are new to you, start with what a limit is.