Limits

A limit describes what a function approaches as its input gets close to a value, without necessarily reaching it. Direct substitution works when the function is defined at that point. If substitution gives $\frac{0}{0}$, simplify first: factor and cancel common terms for polynomials, multiply by the conjugate for square roots, or use standard trigonometric limits such as ${\lim }_{x\to 0}\frac{\sin x}{x}=1$. For limits at infinity of rational functions, divide every term by the highest power of $x$ in the denominator. Once the expression is simplified, substitute again to find the limit.