--- title: "L'Hopital's Rule and Comparing Growth Rates" description: "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." canonical: https://lightmysky.com/learn/mathematics/lhopitals-rule-and-comparing-growth-rates-mt_Ai-A2sPIoE source: https://lightmysky.com/learn/mathematics/lhopitals-rule-and-comparing-growth-rates-mt_Ai-A2sPIoE.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/lhopitals-rule-and-comparing-growth-rates-mt_Ai-A2sPIoE ## Ready when they can - 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 ## Lesson: Settling zero over zero 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. **Example.** 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. **Tip.** 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. **Recap.** Check for 0 over 0 or infinity over infinity, differentiate top and bottom separately, and reshape products before you start. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [The Mean Value Theorem](https://lightmysky.com/learn/mathematics/the-mean-value-theorem-mt_09iApYVelx) - [The Limit Laws and Indeterminate Forms](https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx) ## Opens up - [Asymptotic Notation Made Precise](https://lightmysky.com/learn/computing/asymptotic-notation-made-precise-mt_B0Z6Syy8Ve) - [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z)