Finding limits: graphs, tables and algebra

The limit $\lim_{x \to a} f(x) = L$ means the outputs $f(x)$ get as close to $L$ as you like when $x$ is close enough to $a$, on both sides. The value $f(a)$ plays no part: it can equal $L$, differ from $L$, or not exist at all.

A one-sided limit looks from one side only: $\lim_{x \to a^-}$ uses inputs less than $a$, and $\lim_{x \to a^+}$ uses inputs greater than $a$. The two-sided limit exists exactly when both one-sided limits exist and are equal.

On a graph, trace the curve toward $x = a$ from each side and read the height it heads for. An open circle marks a missing point; a filled dot marks the actual value $f(a)$. In a table, read the outputs for inputs closing in on $a$ from each side. A table only suggests a limit; it cannot prove one.

With a formula, try direct substitution first. Polynomials, roots, exponentials, logarithms, sine and cosine, and rational functions are continuous on their domains, so at an input in the domain the limit is just the value.

If substitution into a quotient gives $\frac{0}{0}$, the form is indeterminate: the algebra is unfinished. Factor and cancel, multiply by a conjugate, or combine fractions, then substitute again. If it gives a nonzero number over 0, the outputs grow without bound near that input and the limit is not a finite number.

Two more tools: $\lim_{x \to 0}\frac{\sin x}{x} = 1$ (radians), and the squeeze theorem: if $g(x) \le f(x) \le h(x)$ near $a$ and $g$ and $h$ have the same limit $L$ at $a$, then so does $f$.