considerando los siguientes limites ,completa las tablas respectivas y estima el limite .compara el resultado con la representacion grafica respectiva
Respuestas a la pregunta
Para resolver éste ejercicio, vamos a calcular los límites de cada uno evaluando el valor de "x".
a. Lim x-> 1 (x²+6x)
X 0.98 0.99 1.01 1.02 1.03
f(x) 6.84 6.91 7.08 7.16 7.24
(x.f(x)) (0.98, 6.84) (0.99, 6.91) (1.01, 7.08) (1.02, 7.16) (1.03, 7.24)
b.- lim x-> 0 (x²+6x/x)
X -0.1 -0.001 0.001 0.1
f(x) 5.9 5.999 6.001 6.1
(x.f(x)) (-0.1, 5.9) (-0.001, 5.99) (0.001, 6.001) (0.1, 6.1)
c.- lim x-> 4 (x²-6x+8)/x-4)
X 3.98 3.99 4.01 4.02 4.03
f(x) 7.92 7.96 8.04 8.08 8.12
(x.f(x)) (3.98, 7.92) (3.99, 7.96) (4.01, 8.04) (4.02, 8.08) (4.03, 8.12)
d.- lim x-> 4 (x²+2x+1)/x+4)
X -0.99 -0.98 1.01 1.02
f(x) 0.00013 0.0005 3.22 3.23
(x.f(x)) (-0.99, 0.00013) (-0.98, 0.0005) (1.01, 3.22) (1.02, 3.23)
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Resolvemos problemas de evaluación de limites:
1.-
Lim x-→ 1 (x²+6x) ≈ 7
para resolver un limite solo debemos evaluar el valor de x al que tiende tal "limite"
- x = 0.98
- f(x) = (0.98² + 6*0.98) = 6.84
x, f(x) = 0.98, 6.84
- x = 0.99
- f(x) = (0.99² + 6*0.99) = 6.91
x, f(x) = 0.99, 6.91
- x = 1.01
- f(x) = (1.01² + 6*1.01) = 7.08
x, f(x) = 1.01, 7.08
- x = 1.02
- f(x) = (1.02² + 6*1.02) = 7.16
x, f(x) = 1.02, 7.16
- x = 1.03
- f(x) = (1.03² + 6*1.03) = 7.24
x, f(x) = 1.03, 7.24
X 0.98 0.99 1.01 1.02 1.03
- f(x) 6.84 6.91 7.08 7.16 7.24
- [x.f(x)] (0.98, 6.84) (0.99, 6.91) (1.01, 7.08) (1.02, 7.16) (1.03, 7.24)
2.- Lim x→0 ((x²+6x)/x) ≈ 6
- x = -0.1
- f(x) = ((-0.1²+6x*-0.1)/x-0.1 = 5.9
x, f(x) = -0.1, 5.9
- x = -0.001
- f(x) = ((-0.001²+6x*-0.001)/x-0.001 = 5.999
x, f(x) = -0.001, 5.999
- x = 0.001
- f(x) = ((-0.001²+6x*-0.001)/x-0.001 = 6.001
x, f(x) = 0.001, 6.001
- x = 0.1
- f(x) = ((0.1²+6x*0.1)/x-0.1 = 6.1
x, f(x) = 0.1, 6.1
X -0.1 -0.001 0.001 0.1
- f(x) 5.9 5.999 6.001 6.1
- [x.f(x)] (-0.1, 5.9) (-0.001, 5.99) (0.001, 6.001) (0.1, 6.1)
3.- lim x → 4 (x²-6x+8)/x-4) ≈ 8
- x = 3.98
- f(x) = (3.98² - 6*3.98 + 8)/(3.98 - 4) = 7.92
x, f(x) = 3.98, 7.92
- x = 3.99
- f(x) = (3.99² - 6*3.99 + 8)/(3.99 - 4) = 7.96
x, f(x) = 3.99, 7.96
- x = 4.01
- f(x) = (4.01² - 6*4.01 + 8)/(4.01 - 4) = 8.04
x, f(x) = 4.01, 8.04
- x = 4.02
- f(x) = (4.02² - 6*4.02 + 8)/(4.02 - 4) = 8.08
x, f(x) = 4.02, 8.08
- x = 4.03
- f(x) = (4.03² - 6*4.03 + 8)/(4.03 - 4) = 8.12
x, f(x) = 4.03, 8.12
X 3.98 3.99 4.01 4.02 4.03
- f(x) 7.92 7.96 8.04 8.08 8.12
- [x.f(x)] (3.98, 7.92) (3.99, 7.96) (4.01, 8.04) (4.02, 8.08) (4.03, 8.12)
4.-
lim x → -1 (x²+2x+1)/x+4) = -2/3
- x = -0.99
- f(x) = (-0.99² + 2*-0.99 + 1)/(-0.99+4) = -0.65
x, f(x) = -0.99, -0.65
- x = -0.98
- f(x) = (-0.98² + 2*-0.98 + 1)/(-0.98+4) = -0.63
x, f(x) = -0.98, -0.63
- x = 1.01
- f(x) = (1.01² + 2*1.01 + 1)/(1.01+4) = 0.80
x, f(x) = 1.01, 0.8
- x = 1.02
- f(x) = (1.02² + 2*1.02 + 1)/(1.02+4) = 0.81
x, f(x) = 1.02, 0.81
X -0.99 -0.98 1.01 1.02
- f(x) -0.65 -0.63 0.80 0.81
- [x.f(x)] (-0.99, -0.65) (-0.98, -0.63) (1.01, 0.8) (1.02, 0.81)
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