Properties and theorems of limits and examples
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This document discusses limit laws and how to evaluate limits. It provides 6 limit laws for evaluating limits of functions, including laws for sums, products, quotients, roots, and composite functions. It also includes an example of applying these laws to evaluate several example limits. Finally, it discusses the squeeze theorem for evaluating limits and provides an example of applying that theorem.
Properties and theorems of limits and examples
- 1. c Amy Austin, September 16, 2015 1 Section 2.3: Limit Laws Limit Laws If lim xa f(x) and lim xa g(x) both exist and c is any constant. Then: (1) lim xa (f(x) 賊 g(x)) = lim xa f(x) 賊 lim xa g(x) (2) lim xa cf(x) = c lim xa f(x) (3) lim xa f(x)g(x) = lim xa f(x) lim xa g(x) (4) lim xa f(x) g(x) = lim xa f(x) lim xa g(x) , provided lim xa g(x) 6= 0. If lim xa g(x) = 0, simplify f(x) g(x) (5) lim xa (f(x))n = lim xa f(x) n (6) lim xa n q f(x) = n q lim xa f(x). In the event n is even, we must have lim xa f(x) 0. EXAMPLE 1: If it is known that lim x2 f(x) = 3, find lim x2 q (f(x))2 + 6 f(x) + 2x + 11 EXAMPLE 2: Find the limit. If the limit does not exist, explain why. (a) lim x1 x4 + x2 6 x4 + 2x + 3 (b) lim x3 x2 x 12 x + 3
- 2. c Amy Austin, September 16, 2015 2 (c) lim x9 x2 81 x 3 (d) lim x6 1 6 1 x x 6 (e) Find lim x5 f(x), lim x1 f(x), and lim x1 f(x) where f(x) = 錚 錚 錚 錚 錚 錚 錚 4 + 5x if x 1 x if 1 x 1 3 if x = 1 4 x if x 1 .
- 3. c Amy Austin, September 16, 2015 3 (f) Find lim x2 |x 2| x 2 (g) Let f(x) = x2 + 3x |x + 3| . Find lim x3 f(x) and lim x3+ f(x). Squeeze (aka sandwich) Theorem If f(x) g(x) h(x) for all x in an interval containing x = a (except possibly at x = a) and lim xa f(x) = lim xa h(x) = L, then lim xa g(x) = L. EXAMPLE 3: Given 3x f(x) x3 +2 for all x in the interval (0, 3), find lim x1 f(x).