L’Hôpital’s rule & indeterminate forms
L’Hôpital’s rule says that for a 0/0 or ∞/∞ limit, lim f/g = lim f′/g′ provided the second limit exists. The rule applies to exactly those two forms — applying it to a limit that is not indeterminate produces wrong answers — and the other indeterminate forms (0·∞, ∞−∞, 1^∞, 0⁰, ∞⁰) must first be rewritten as a quotient.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Verify the form is 0/0 or ∞/∞Substitute first. Anything else must be converted before the rule applies.
- Differentiate numerator and denominator SEPARATELYThis is not the quotient rule — you differentiate top and bottom independently.
- Repeat if still indeterminateThe rule can be applied several times, checking the form each round.
- Convert other forms0·∞ becomes a quotient by moving one factor to the denominator; 1^∞ and 0⁰ are handled by taking logarithms first.
Worked example
Evaluate lim_{x→0} (eˣ − 1 − x)/x².
- Substitute: (1 − 1 − 0)/0 = 0/0 ✓ indeterminate.
- Differentiate top and bottom: (eˣ − 1)/(2x). At x = 0 this is again 0/0.
- Apply again: eˣ/2.
- Substitute x = 0: 1/2.
Answer. 1/2 — confirmed by the Taylor series eˣ − 1 − x = x²/2 + O(x³).
Where marks get dropped
These are the specific errors that cost credit on l’hôpital’s rule & indeterminate forms questions — QED's rubric penalises each of them separately.
- Applying the quotient rule instead of differentiating separately. L’Hôpital differentiates numerator and denominator independently.
- Using the rule on a determinate form. lim_{x→0}(x+1)/(x+2) = 1/2 directly; L’Hôpital would wrongly give 1.
- Continuing to apply the rule after the form stops being indeterminate, which produces a different and wrong answer.
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L’Hôpital’s rule & indeterminate forms — frequently asked questions
How do I handle 0·∞?
Rewrite as a quotient: f·g = f/(1/g). For x ln x as x → 0⁺, write it as ln x/(1/x), which is ∞/∞ and yields 0.
How do I handle 1^∞?
Take logs. Let L be the limit, compute ln L as a 0·∞ limit, then exponentiate. This is how lim (1+1/n)ⁿ = e is derived.
Can I use Taylor series instead?
Often yes, and it is frequently faster and more reliable. Expanding numerator and denominator to a few terms shows the leading behaviour immediately.
The rest of Calculus
Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Limits & continuity
- 2Product, quotient & chain rules
- 3Tangent lines & linear approximation
- 4Extrema & optimisation
- 5Indefinite & definite integrals
- 6Substitution & integration by parts
- 7Series & convergence basics
- 8Implicit differentiation & related rates
- 9Mean value & intermediate value theorems
- 10Improper integrals & convergence
- 11Taylor & Maclaurin polynomials
- 12L’Hôpital’s rule & indeterminate forms
- 13Curve sketching: asymptotes, concavity & inflection
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