--- title: "The Limit Laws and Indeterminate Forms" description: "Apply finite limit laws where their conditions hold. Investigate 0/0 forms with suitable algebra or bounds; the form alone guarantees neither a removable factor nor a finite limit." canonical: https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx source: https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx.md retrieved: 2026-11-12 --- > **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`. # The Limit Laws and Indeterminate Forms Apply finite limit laws where their conditions hold. Investigate 0/0 forms with suitable algebra or bounds; the form alone guarantees neither a removable factor nor a finite limit. Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx ## Ready when they can - Apply finite limit laws and investigate indeterminate quotients using valid algebra - Evaluate root-difference limits by rationalising and checking the remaining expression - Use bounding arguments and radian trigonometric limits, including rescaling ## Lesson: Taming zero over zero If f and g have finite limits L and M, their sum has limit L+M and their product has limit LM. Their quotient has limit L/M only when M is nonzero. Constants keep their values. For example, if the component limits are 2 and 4, adding one to their product gives limit 2(4)+1 = 9. A 0/0 result from substitution is indeterminate. It does not decide whether the limit is finite, infinite or absent. A counterexample to automatic nonexistence is x/x: away from zero it equals 1. A common factor need not remove every singularity, as x/x² leaves 1/x. **Example.** For (x²-25)/(x-5) near 5, factor the numerator as (x-5)(x+5). Cancel only for x ≠ 5. The nearby expression becomes x+5 and has limit 10. Equal expressions on a punctured neighborhood have the same limit; the expression need not be defined at the approach point. For cubes, use x³-a³ = (x-a)(x²+ax+a²) and x³+a³ = (x+a)(x²-ax+a²). If the denominator contains the matching linear factor, cancel it where nonzero. Evaluate the remaining polynomial at the approach point. Check what remains instead of assuming cancellation guarantees a finite limit. **Example.** For (sqrt(x+16)-4)/x near 0, multiply top and bottom by sqrt(x+16)+4. The numerator becomes x. For x ≠ 0, cancellation leaves 1/(sqrt(x+16)+4), whose limit is 1/8. The identity (a-b)(a+b) = a²-b² also works when a root difference is in the denominator. The squeeze theorem requires lower and upper bounds near the point with the same limit. If A-(x-c)² ≤ g(x) ≤ A+(x-c)², then g tends to A at c. Bounds with different limits do not establish a unique limit. For instance, bounds -2 and 2 alone allow several constant functions. A bounded factor can be controlled by a vanishing amplitude. If |b(x)| ≤ M near c for a fixed M ≥ 0 and a(x) tends to 0, then |a(x)b(x)| ≤ M|a(x)| tends to 0. The product therefore tends to 0. Sine is bounded by 1 in magnitude, even when its argument keeps changing. Adding a constant shifts the product's limit by that constant. The bounds establish the result without substituting into an oscillating factor. Choose a method from the actual expression, not from 0/0 alone. Factoring or conjugates may give an equivalent expression; inspect its remaining limit. Bounds may establish a limit directly. Neither cancellation nor squeezing promises that final substitution into the original expression will work. A radian angle is signed arc length divided by radius. On the unit circle, cos u and sin u are the horizontal and vertical coordinates; tan u = sin u/cos u where cos u ≠ 0. Cosine is continuous and cos 0 = 1. For 0 < |u| < pi/2, cos u ≤ sin u/u ≤ 1, so sin u/u tends to 1. For fixed k ≠ 0, sin(kx)/x = k sin(kx)/(kx) tends to k. Also tan(kx)/x = [sin(kx)/(kx)][k/cos(kx)] tends to k. *(drawing: Algebra and bounds are different arguments. Each needs its own conditions.)* **Recap.** A 0/0 form needs investigation. Valid algebra can preserve nearby values; matching bounds can establish a limit directly. *Original limit-law lesson, questions and flashcards written for this learning dataset.* ## Practice 20 questions on this page, each with its working shown. ## Needs first - [One-Sided Limits and When a Limit Fails to Exist](https://lightmysky.com/learn/mathematics/one-sided-limits-and-when-a-limit-fails-to-exist-mt__L0B_WQq-P) ## Optional connections - [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C) ## Opens up - [L'Hopital's Rule and Comparing Growth Rates](https://lightmysky.com/learn/mathematics/lhopitals-rule-and-comparing-growth-rates-mt_Ai-A2sPIoE) - [Continuity and the Intermediate Value Theorem](https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV)