Limits at Infinity

Business Calculus · Axiom Academy

Understanding long-term behavior of functions In business, we often need to know: "What happens as production grows very large?" or "Where does this trend stabilize?" These are questions about limits at infinity . _ x f(x) = L means that as x grows larger and larger, f(x) approaches the value L. _ x - f(x) = L means the same as x becomes more and more negative. Business Example: Average Cost A factory's average cost is (x) = x + 5 As production increases indefinitely, average cost approaches 5 per unit (the variable cost). Fixed costs become negligible! Here are the building blocks for evaluating limits at infinity: Constants don't change as x grows. Powers of x grow without bound. Fractions with x in the denominator approach zero. Rule 3 is the most useful! When evaluating rational functions at infinity, terms with x in the denominator "vanish" (approach 0). Rational Functions at Infinity For a rational function q(x) , compare the highest powers (degrees) of the numerator and denominator: Exponential and Logarithmic Limits Exponential functions dominate polynomial functions as x → ∞: Business Application: Learning Curve A worker's production rate approaches a maximum: P(t) = 100(1 - e^ -0.1t ) Production approaches 100 units/hour as experience increases. When _ x f(x) = L or _ x - f(x) = L , the line y = L is a horizontal asymptote . Evaluate both _ x f(x) and _ x - f(x) . A function can have: No horizontal asymptotes (polynomial) One horizontal asymptote (exponential decay)

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