limit as x approaches 1 of [ (x^5 - 1) / (x^3 - 1)]?

Question

Determine whether the series converges or diverges?

Answer 1

Use L'Hospital's Rule

lim[x -> 1]{(x^5-1)/(x^3-1)} = lim(x ->1){5x^4/3x^2) =lim[x -> 1] 5x^2/4 = 5/4

Answer 2

lim x->1 (x^5 - 1) / (x^3 - 1)
= lim x->1 (x-1) ( x^4 + x^3 + x^2 + x + 1)/ (x-1) (x^2 + x + 1)
= lim x->1 ( x^4 + x^3 + x^2 + x + 1)/ (x^2 + x + 1)
= (1+1+1+1+1)/ (1+1+1) = 5/3