--- title: "One-Sided Limits and When a Limit Fails to Exist" description: "Treat a limit as a two-sided agreement: what the outputs approach from the left has to match what they approach from the right. Where the two disagree, or the outputs grow without bound, no limit exis" canonical: https://lightmysky.com/learn/mathematics/one-sided-limits-and-when-a-limit-fails-to-exist-mt__L0B_WQq-P source: https://lightmysky.com/learn/mathematics/one-sided-limits-and-when-a-limit-fails-to-exist-mt__L0B_WQq-P.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`. # One-Sided Limits and When a Limit Fails to Exist Treat a limit as a two-sided agreement: what the outputs approach from the left has to match what they approach from the right. Where the two disagree, or the outputs grow without bound, no limit exists. Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/one-sided-limits-and-when-a-limit-fails-to-exist-mt__L0B_WQq-P ## Ready when they can - Read the left and right limits off a piecewise graph at a jump - Give a function whose limit at a point fails because the one-sided values differ - Separate a limit that is infinite from one that fails to exist at all ## Lesson: Both sides must agree A limit is a two-sided agreement. As x walks toward a number c, the outputs must settle on the same target whether x comes from the left or from the right. So you always run the same check: find the left-hand limit, find the right-hand limit, then compare. If the two match, the limit exists and equals that shared value. If they differ, no two-sided limit exists. **Example.** Try f(x) = |x| / x at 0. For positive x the fraction equals 1, so the right-hand limit is 1. For negative x it equals -1, so the left-hand limit is -1. The values 1 and -1 disagree, so the limit at 0 does not exist, even though each side alone behaves perfectly. Piecewise graphs work the same way. Take f(x) = x + 3 below 1 and f(x) = 5 - x above 1. The left side gives 1 + 3 = 4 and the right side gives 5 - 1 = 4. The sides agree, so the limit at 1 is 4. Unbounded growth needs the same comparison. For f(x) = 1 / (x - 3) squared, the denominator shrinks to zero but stays positive on both sides, so the outputs grow without bound in one shared direction, and we call the limit infinite. For f(x) = 1 / x at 0 the left side dives toward negative infinity while the right side climbs toward positive infinity, so the sides disagree and the limit does not exist. The same verdict holds when only one side misbehaves: if the curve shoots upward without bound from the left of x = 4 but settles at height -2 from the right, there is no shared target, so the limit does not exist. **Tip.** Read every graph with one habit. Trace the curve in from the left and record the height. Trace it in from the right and record that height. Then compare. Matching heights give the limit. Different heights, or sides racing off in opposite directions, mean no limit. Never average the two sides: for f(x) = 2x + 1 below 2 and 8 - x above, the sides give 5 and 6, and 5.5 is not the answer. The answer is that no limit exists. **Recap.** Check the left, check the right, compare: a shared target is a limit, anything else is not. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [The Gradient of a Curve as a Limit](https://lightmysky.com/learn/mathematics/the-gradient-of-a-curve-as-a-limit-mt_LAlhcwVjln) - [Differentiation from First Principles](https://lightmysky.com/learn/mathematics/differentiation-from-first-principles-mt_pdfPztEjw_) ## Opens up - [The Limit Laws and Indeterminate Forms](https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx)