This isn't magic, and it isn't a formula to take on faith. It falls right out of the last lesson — local linearity — in five short moves.
Substitute and you get 0/0 — two things both racing to zero, and it looks hopeless. So dive in. Fall toward the point where they meet, and watch what happens: the bendy curves straighten into lines. That's not a trick — get close enough to any smooth curve and it becomes its own tangent line. And once both are just lines, the fraction that looked like nothing is only ever slope ÷ slope. Pick a limit, then hit dive.
Four moves, every time. Miss the first and everything after it is wrong — you can't apply L’Hôpital to a limit that isn't indeterminate.
Substitute. Only if you get 0/0 or ±∞/∞ may you continue. Any other value — you're already done; just report it.
Separately. Not the quotient rule — take f′ and g′ on their own and form the new ratio f′/g′.
Substitute again into f′/g′. If it resolves, that's your answer.
If f′/g′ is also indeterminate, apply the rule again — and again — until it resolves.
Only 0/0 and ∞/∞ feed L’Hôpital directly. The other five are indeterminate too, but you must first massage them into one of those — usually with algebra or a logarithm.
L’Hôpital is only valid for 0/0 or ∞/∞. Applying it to a determinate limit gives a confident, wrong answer.
Differentiate the top and the bottom separately. It looks like the quotient rule but it isn't — no product or square in sight.
Near a, f and g are their tangent lines, so their ratio is the ratio of slopes. The (x−a) that made both vanish cancels out.
After one round, look again: it may resolve, or need another pass. Stop the instant it's no longer indeterminate — extra passes give wrong answers.
Six limits — including one that looks like a job for L’Hôpital but isn't. Check the form, then decide. Try each before revealing.
Six fast checks. Pick an answer to see whether it's right and why — then work all six.
The cardinal sin. lim (x+1)/(x+3) at 0 is 1/3 — done. Differentiating gives 1/1 = 1, a confident wrong answer. Always substitute first.
L’Hôpital wants f′/g′ — the top's derivative over the bottom's derivative, computed separately. The quotient rule is a different (wrong here) beast.
One pass may leave you at 0/0 again (apply again) — or at a clean value (stop). Blindly differentiating a resolved limit re-breaks it.
Those are indeterminate but not fractions yet. Rewrite as a single 0/0 or ∞/∞ quotient before the rule applies.
Power forms need a logarithm first: take ln, evaluate that limit with L’Hôpital, then exponentiate the result back.
Check the form in the direction the limit is taken (x→0⁺, x→∞). The rule works for ∞/∞ too — don't restrict it to 0/0.