limit existing evaluate

Question

Evaluate limit if it exists

lim x+1 x→−1. (x+1/ x^3+1)

Answer 1

Write x^3+1=(x+1)(x^2-x+1) so that (x+1)/(x^3+1)=1/(x^2-x+1). Hence the limit of the ratio (x+1)/(x^3-x^2+1) as x->-1 is equal to (1/((-1)^2-(-1)+1)=1/3.

Answer 2

For x≠-1 (x+1)/(x³+1) = 1/(x²-x+1) (You get this by factoring or, if necessary, by long division)

Consequently the limit is 1/3.