limits calculus

Question

lim      of   8+ x - 4x²       

x→∞        √6+ x² + 7x⁴

Answer 1

lim_x→∞ [ (8 + x - 4x²) / √ (6 + x² + 7x⁴)] 

 

= lim_x→∞ [ (x²( -4 + 1/x + 8/x²} / {√(x⁴)√(6/x⁴ + 1/x² + 7)}]

 

= lim_x→∞ [ (-4 + 1/x - 4x²) / √(6/x⁴ + 1/x² + 7) ]

 

= (-4 + 0 - 0) / √ (0 + 0 + 7) = -4/√7 = -4√7 / 7

 

Answer 2

Are you familiar with L'Hopital's rule? If the limit as x tends to infinity (Lim x ->∞) of f(x)/g(x) = ∞/∞, then Lim x ->∞ f(x)/g(x) = Lim x ->∞ f'(x)/g'(x) where f'(x) is the derivative of f with respect to x.