Determine o lim x->∞ (raiz(x^2 + 1) - raiz(x))

Passo 1

Aplicamos a regra de L'Hôpital, multiplicando inicialmente pelo termo conjugado de x 2 + 1 - x que e x 2 + 1 + x . Assim:

l i m x → ∞ x 2 + 1 - x = l i m x → ∞ x 2 + 1 - x ⋅ x 2 + 1 + x x 2 + 1 + x = l i m x → ∞ x 2 - x + 1 x 2 + 1 + x =  

l i m x → ∞ x 2 - x + 1 x 2 + 1 + x = l i m x → ∞ d d x x 2 - x + 1 d d x x 2 + 1 + d d x x = l i m x → ∞ 2 x - 1 1 2 2 x x 2 + x - 1 2 1 x = l i m x → ∞ 2 x - 1 1 2 2 x x 1 + 1 / x - 1 2 1 x

l i m x → ∞ 2 x - 1 1 2 2 x x 1 + 1 / x - 1 2 1 x = l i m x → ∞ 2 x - 1 1 2 2 1 - 1 2 1 x = l i m x → ∞ 2 x - 1 2 - 0 = ∞

Resposta

l i m x → ∞ x 2 + 1 - x = ∞

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