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Calc 2.3 - Limit Laws

IF c is constant and lim f(x) and lim g(x) exist then…. x->a. x->a. 1. lim [f(x) + g(x)] = lim f(x) + lim g(x). x->a. x->a. x->a. 2. lim [cf(x)] = c lim f(x). x->a. x->a. 3. lim [f(x)g(x)] = lim f(x) • lim g(x). x->a. x->a. x->a. f(x). lim f(x). 4. lim. =. x->a. g(x). x->a.

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Calc 2.3 - Limit Laws

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  1. IF c is constant and lim f(x) and lim g(x) exist then… x->a x->a 1. lim [f(x) + g(x)] = lim f(x) + lim g(x) x->a x->a x->a 2. lim [cf(x)] = c lim f(x) x->a x->a 3. lim [f(x)g(x)] = lim f(x) • lim g(x) x->a x->a x->a f(x) lim f(x) 4. lim = x->a g(x) x->a (= 0) lim g(x) x->a Calc 2.3 - Limit Laws “The limit of a sum (or diff) is the sum (or diff) of the limits” “The limit of a constant times a function is the constant times the limit of the function.” “The limit of a product is the product of the limits” “The limit of a quotient is the quotient of the limits”

  2. a) lim [f(x) + 5g(x)] f x->-2 b) lim [f(x)g(x)] x->1 g f(x) c) lim g(x) x->2 ex: Use the limit laws and the graphs of f and g below to evaluate the following limits if they exist…

  3. n n lim [f(x)] = lim f(x) x->a x->a n n n n lim c = c lim x = a lim x = a lim x = a x->a x->a x->a x->a n n 7. Root Law: lim f(x) = lim f(x) (where n is a positive integer) x->a x->a Additional Laws 5. Power Law: 6. Special Limits:

  4. lim f(x) = f(a) x->a ex: lim 2x2 - 3x + 4 = [2(5)2 - 3(5) + 4] = 39 x->5 “Continuous at a” (x + 1)(x - 1) x2 - 1 lim lim lim (x + 1) = 2 = x - 1 x - 1 x->1 x->1 x->1 “If f is a polynomial or rational function and a is in the domain of f then”:… Direct Substitution =

  5. (3 + h)2 - 9 ex: Evaluate lim h-->0 h - 9 6h + h2 h (6 + h) lim lim lim (6 + h) = 6 = = = h-->0 h h-->0 h h-->0 t2 + 9 - 3 t2 + 9 + 3 (t2 + 9) - 9 ex: Find lim lim = t-->0 t2 t-->0 t2 + 9 + 3 t2 t2 + 9 + 3 t2 1 1 1 lim lim lim = = = = t-->0 t-->0 t-->0 6 9 + 3 t2 t2 + 9 + 3 t2 + 9 + 3 (3 + h)2 = (3 + h)(3 + h) = [9 + 6h + h2]

  6. If f(x) < g(x) when x is near a (except possibly at a), and the limits of f and g both exist as x approaches a, then: lim f(x) < lim g(x) x-->a x-->a The Squeeze Theorem If f(x) < g(x) < h(x) when x is near a (except possibly at a) and: lim f(x) = lim h(x) = L x-->a x-->a then… lim g(x) = L x-->a

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