Question

[(x² + 5x + 3)/(x² + x + 2)]x for limx→∞ = ? 

(a) e^–4 

(b) e² 

(c) e⁴ 

(d) e^–2

Answer 1

The correct option (c) e⁴

Explanation:

lim_x→∞ [(x² + 5x + 3)/(x² + x + 2)]^x

= lim_x→∞ [{(x² + x + 2) + (4x + 1)}/(x² + x + 2)]^x

= lim_x→∞ [1 + {(4x + 1)/(x² + x + 2)}]^x

= lim_x→∞ {[1 + {(4x + 1)/(x² + x + 2)}]^{[{(x)2+(x)+(2)}/(4x+1)]}}^α (1)

where

α = [{x(4x + 1)}/(x² + x + 2)] = [{4 + (1/x)}/{1 + (1/x) + (2/x²)}]

as x → ∞, α → 4

∴ from (1), we can write,

lim_x→∞ [(x² + 5x + 3)/(x² + x + 2)]^x = e⁴.