lim (-2x+1)/?(x^2 +x) is indeterminate (?/?).
So we use an algebraic manipulation to find the limit.
= lim [(-2x+1)/?(x^2 +x)] * [(1/?x^2) / (1/?x^2)]
= lim [(-2x+1) * (1/?x^2)] / [?(x^2 +x) * (1/?x^2)]
= lim [(-2x+1) * (1/|x|)] / [?(1 + 1/x)]
Since x approaches -?, |x| = -x.
= lim [(-2x+1) * (1/-x)] / [?(1 + 1/x)]
= lim [2-1/x] / [?(1 + 1/x)]
= 2 / ?1 = 2
The limit is 2.