Exercícios Limites - Cálculo 1 2023-1

Ex.: Calcule lim x→1 (x^3 + 1)/(x^2 + 4x + 3) lim x→1 (x^3 + 1)/(x^2 + 4x + 3) = 0/0 divisao de polinomio lim x→1 (x^3 + 1)/(x^2 + 4x + 3) = lim x→1 ( (x+1)(x^2-x+1) )/( (x+1)(x+3) ) = 3/2 Ex.: Calcule lim x→2 (³√x - ³√2)/(x-2) lim x→2 (³√x - ³√2)/(x-2) = 0/0 (indet.) (³√x)^3 = (x^(1/3))^3 = x^(3/3) = x x - 2 = (³√x)^3 - (³√2)^3 = (³√x - ³√2)(³√x^2 + ³√x² + ³√4) a^3 - b^3 = (a - b)(a^2 + ab + b^2) a = ³√x b = ³√2 → ³√x² ³√2 x ³√x = ³√x . x = ³√x² → ³√2 . x = ³√2 . 2 = ³√4 lim x→2 (³√x - ³√2)/(x-2) = lim x→2 (³√x - ³√2)/(³√x - ³√2)(³√x² + ³√x^2 + ³√4) = 1/³√4 → ³√2 . 2 = ³√4 Ex.: Determine graficamente os limites abaixo a) lim x→2 x^2 b) lim x→1 f(x) ; onde f(x) = { x ; se x ≤ 1 x^2 + x - 1 ; se x > 1 a) f(x) = x ; x ≤ 1 f(x) = x^2 + x - 1 ; x > 1 D(f) = {x ∈ ℝ , x ≤ 1} Im(f) = (-∞,0] ∪ [0,1] lim x→1 lim x→1 x² + x - 1 = 1 f(x) = x^2 + x - 1 ; x > 1 D(f) = ℝ Im(f) = [1, +∞)