Limit Notation

Business Calculus · Axiom Academy

Learn to read, write, and interpret limit expressions fluently Limit notation packs a lot of meaning into a compact form. Let's break it down: Read as: "The limit of 2x + 1 as x approaches 3 equals 7." Meaning: When x gets very close to 3, the value of 2x + 1 gets very close to 7. Sometimes we want to specify which direction x is approaching from. This gives us one-sided limits : x approaches a from values less than a The minus superscript (⁻) means "from the left" x approaches a from values greater than a The plus superscript (⁺) means "from the right" The two-sided limit _ x a f(x) = L exists if and only if: Both one-sided limits must exist and be equal! There are two different ways infinity can appear in limit notation: When we write _ x a f(x) = , the limit technically does not exist (since ∞ is not a real number). We use the notation to describe the function's behavior. Business Application: Long-Run Average Cost If average cost is (x) = x + 12 , then: As production increases indefinitely, average cost approaches 12 (the variable cost per unit). Practice Reading Limit Notation Being able to read limit expressions fluently is essential. Here's a reference: The arrow (→) is always read as "approaches" or "goes to." The subscript below "lim" tells you what is approaching what value . A limit _ x a f(x) = L exists when: Both one-sided limits are equal to L

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