Verified answer

x³ + y³ = (x + y)(x² - xy + y²)

1. lim(x->-1) (x³ + 1)/(x² + 3x + 2) = lim(x->-1) (x + 1)(x² - x + 1)/(x+ 1)(x + 2) =  lim(x->-1) (x + 1)(x² - x + 1)/(x+ 1)(x + 2) = ((-1)² - (-1) + 1)/(-1 + 2) = 3

2. lim(x->-2) (x³ + 1)/(x² + 3x + 2) = lim(x->-) (x³ + 1)/(x+ 1)(x + 2) = ((-2)³ +1)/(-1)*0 = -7/0 = -∞

3. вертикальная асимптота  х = -2

4. наклонная y(x) = kx + b

k = lim(x->∞) y(x)/x

b = lim(x->∞) (y(x) - kx)

k = lim(x->∞) y(x)/x = lim(x->∞) (x³ + 1)/x(x² + 3x + 2) = lim(x->∞)(1 + 1/x³)/(1 + 3/x + 2/x²) = (1 + 0)/(1 + 0 + 0) = 1

b = lim(x->∞) (y(x) - kx) = lim(x->∞) [(x³ + 1)/(x² + 3x + 2) - x] = lim(x->∞) (x³ + 1 - x³ - 3x² - 2)/(x² + 3x + 2)  = lim(x->∞) ( - 1  - 3x² )/(x² + 3x + 2)  = lim(x->∞) ( - 1/x²  - 3 )/(1 + 3/x + 2/x²)  = (0 - 3)/(1 + 0 + 0) = -3

y = x - 3