L'Hopital's Rule and Comparing Growth Rates
Resolve 0/0 and infinity over infinity limits by differentiating numerator and denominator separately, and use the result to rank how fast logarithms, powers and exponentials grow.
What a learner can do afterwards
- Apply the rule twice on a limit that stays indeterminate after one pass
- Rewrite a product or power indeterminate form so the rule applies
- Rank ln x, x^k and e^x by growth and support the ranking with a limit
1 · Read
Some limits arrive as standoffs: top and bottom both die to zero, or both run to infinity. When that happens you may differentiate the top and the bottom separately and take the limit again. That is the whole rule, and it never differentiates the quotient as one piece. The classic sine of x over x tends to 0 over 0, and the rule gives 1.
Take 1 minus cosine x, over x squared, as x tends to 0. Direct substitution gives 0 over 0, so round one gives sine x over 2 x, still 0 over 0. Round two gives cosine x over 2. At zero that is 1 over 2, so the limit is 0.5.
Products need reshaping first, since the rule only reads quotients. Write x times ln x as ln x over 1 over x: same value, but 0 times negative infinity became negative infinity over infinity. The rule then drives it to 0. Always check the form before every round, and stop the moment it breaks: using the rule on a settled form makes wrong answers.
Limits at infinity rank growth into strict leagues. Logs crawl below every power, and every fixed power falls to the exponential. So x squared over e to the x tends to 0, and even x to the 100 over e to the x tends to 0. Vote with the dominant term first, then verify with the rule.
The rule runs on derivatives, so know two cold: e to the x reproduces itself, which is why exponentials survive round after round, and ln x gives 1 over x, which is why logs collapse fast. Each round is legal only while 0 over 0 or infinity over infinity lasts.
Check for 0 over 0 or infinity over infinity, differentiate top and bottom separately, and reshape products before you start.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.