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

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

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

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

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