f(t) = t * i + 2 * t² * j + 3 * t³ * k

g(t) = 2t i + j - 3t^2 k, t ≤ 0

Passo 1

  1. underset(t -> 1)(lim) [ vec f (t) + vec g (t) ]

Pessoal, antes de tudo, vamos calcular vec f (t) + vec g ( t ) :

vec f (t) + vec g (t) = (t vec i + 2 t^2 vec j + 3 t^3 vec k) + (2 t vec i + vec j - 3 t^2 vec k) = (t + 2 t) vec i + (2 t^2 + 1) vec j + (3 t^3 - 3 t^2) vec k = 3 t vec i + (2 t^2 + 1) vec j + (3 t^3 - 3 t^2) vec k

Agora podemos aplicar o limite!

underset(t -> 1)(lim) [vec f (t) + vec g (t)] = underset(t -> 1)(lim) ( 3 t ) vec i + underset(t -> 1)(l i m) (2 t^2 + 1) vec j + underset(t -> 1)(l i m) (3 t^3 - 3 t^2) vec k = 3 vec i + (2 + 1) vec j + (3 - 3) vec k

underset(t -> 1)(l i m) [vec f (t) + vec g (t)] = 3 vec i + 3 vec j + 0 vec k

Resposta

  1. underset(t -> 1)(l i m) [vec f (t) + vec g (t)] = 3 vec i + 3 vec j