ليكن التابع المعرف على IR وفق: F(x) = (x2+2x+2).e^-x احسب Lim A(α) α→+∞

F(x) = (x2+2x+2).e^-x

Lim f(x) x→-∞ = Lim [x2.e^-x] x→-∞ = +∞

Lim f(x) x→+∞ = Lim [x2/e^x + 2x/e^x + 2/e^x] x→+∞

= Lim [x2/e^x] x→+∞ + Lim [2x/e^x] x→+∞ + Lim [2/e^x] x→+∞

= 0 + 0 + 0 = 0

المستقيم y=0 وهو محور الفواصل مقارب باتجاه +∞

F’(x) = (2x+2).e^-x – e^-x(x2 + 2x + 2) = -x2.e^-x < 0

F’(x) = 0

X = 0

F(0) = 2

X = -1

F(-1) = e

F’(-1) = -e

y = f’(-1)(x+1) + f(-1)

y = -e(x+1) + e

y = -c.x

اذا كان F تابعاً اصلياً للتابع f

F’(x) = f(x)

(2ax+b).e^-x – e^-x(ax2+bx+c) = (x2+2x+2).e^-x

-ax2 + (2a-b)x + (b-c) = x2+2x+2

b-c = 2

2a-b = 2

-a = 1

C = -6

b = -4

a = -1

f(x) = (-x2-4x-6).e^-x

A(α) = 〖[(-x^2  – 4x – 6).e^(-x)]〗_0^∝

= - (α2 - 4α - 6).e^-α + 6

Lim A(α) α→+∞ = Lim [-α2/e^α - 2α/e^α – 6/e^α + 6] α→+∞

= (0-0-0) + 6

Lim A(α) α→+∞ = 6