Step by Step
L
L'Hôpital's Rule itself
If a limit of f(x)/g(x) evaluates to the indeterminate form 0/0 or ∞/∞, you can differentiate the numerator and denominator separately (not using the quotient rule — just their individual derivatives) and take the limit of that new ratio instead.
I
"If" — always check the form first
L'Hôpital's Rule can only be applied when the limit is genuinely 0/0 or ∞/∞. Applying it to a limit that isn't actually in one of these indeterminate forms will give you a wrong answer.
P
"Problem" is the indeterminate form
This indeterminate form is exactly what makes the original limit impossible to evaluate by direct substitution — that's the "problem" the rule solves.
S
"Separately" differentiate top and bottom
Differentiate the numerator and denominator independently of each other — do NOT use the quotient rule on the fraction as a whole. If the new limit is still indeterminate, you can apply L'Hôpital's Rule again, repeating as needed.
Applied Walkthrough
1
Evaluate lim(x→0) sin(x)/x. Direct substitution gives sin(0)/0 = 0/0 — an indeterminate form, so L'Hôpital's Rule applies.
2
Differentiate the numerator and denominator separately: numerator becomes cos(x), denominator becomes 1.
3
Take the limit of the new ratio: lim(x→0) cos(x)/1 = cos(0)/1 = 1.
4
So the original limit is 1 — a classic result confirming that sin(x)/x approaches 1 as x approaches 0.
Exam Application
Exams test whether you correctly verify the limit is genuinely 0/0 or ∞/∞ before applying the rule, and whether you differentiate the numerator and denominator separately rather than using the quotient rule.
⚠ Common Trap
The most common trap is applying L'Hôpital's Rule without checking that the limit is actually an indeterminate 0/0 or ∞/∞ form first — using it on a limit that isn't indeterminate gives an incorrect result.
✓ Quick Self-Check
1. When can L'Hôpital's Rule be applied?
Only when a limit evaluates to the indeterminate form 0/0 or ∞/∞.
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2. How do you differentiate the numerator and denominator when applying L'Hôpital's Rule?
Separately and independently — not using the quotient rule on the whole fraction.
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3. Evaluate lim(x→0) sin(x)/x using L'Hôpital's Rule.
1.
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4. What should you do if, after applying L'Hôpital's Rule once, the limit is still indeterminate?
Apply the rule again, repeating as needed.
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5. What is the biggest risk of applying L'Hôpital's Rule incorrectly?
Applying it to a limit that isn't actually in an indeterminate 0/0 or ∞/∞ form, which gives a wrong answer.
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