Concept preview

What you will be able to do

1. 01Evaluate limits of sums, differences, products, quotients, constant multiples, and powers by applying the limit laws one step at a time.
2. 02Use direct substitution to evaluate the limit of any function that is continuous at the target point, and explain why substitution is legitimate there.
3. 03Recognize the single precondition that makes the quotient law fail — a denominator whose limit is zero — and know that this signals a switch to algebraic techniques.

Every calculus idea that follows — derivatives, integrals, series — is built on limits. So the practical question is: given a formula, how do I actually compute $\lim_{x\to c} f(x)$ quickly and correctly? The answer for most well-behaved functions is a small toolkit of algebraic rules called the limit laws, plus one shortcut called direct substitution. Master these and you can evaluate the vast majority of limits in a single confident line.