Limit Laws and L'Hopital's Rule

Most limits never need the epsilon-delta definition. They fall to a handful of rules plus one theorem.

The limit laws

If $\lim_{x\to a} f(x) = L$ and $\lim_{x\to a} g(x) = M$, both existing:

For polynomials and rational functions with non-zero denominator, this reduces to substitution: put $x = a$ in and evaluate.

Where substitution fails

Substitution breaks when it produces something meaningless:

These are the indeterminate forms. Indeterminate does not mean "no limit" — it means this method cannot tell you. The limit may exist, may be anything at all, and needs a different technique.

Contrast $\dfrac{5}{0}$, which is not indeterminate: it tells you the function grows without bound.

Technique 1: factor and cancel

Substitution gives $0/0$. But for $x \neq 1$:

The cancellation is legal precisely because the limit never evaluates at $x = 1$ — the same exclusion from the definition, doing useful work.

Technique 2: L'Hopital's rule

If $\lim_{x\to a} f(x) = \lim_{x\to a} g(x) = 0$, or both are $\pm\infty$, and $g'(x) \neq 0$ near $a$, then

provided the right-hand limit exists.

Differentiate the numerator and denominator separately. It is not the quotient rule and looks nothing like it.

Three standard limits worth memorising

Check the third with L'Hopital, applied twice:

Each application needs its own $0/0$ check. The second step is legal only because $\sin x \to 0$ and $2x \to 0$.

The mistake examiners look for

Applying L'Hopital when the form is not indeterminate.

Misapplying L'Hopital gives $\lim \frac{1}{1} = 1$ — wrong. The rule is not a general shortcut for quotients; it is a theorem with a hypothesis. Check the form first, every time.