Calculus of a Single Variable, Chapter 3, 3.5, Section 3.5, Problem 28

You need to evaluate the limit, hence, you need to replace oo for x in equation:
lim_(x->-oo) x/(sqrt(x^2+x))= (-oo)/ (sqrt(oo+(-oo)))
Since the result is indeterminate, you need to factor out x^2 at denominator:
lim_(x->-oo) x/(sqrt (x^2(1 +1/x))) =
lim_(x->-oo) x/(|x|sqrt (1 + 1/x)) = lim_(x->-oo) x/(-x*sqrt (1 + 1/x))
Since lim_(x->-oo) 1/x = 0 , yields:
lim_(x->-oo) x/(-x*sqrt (1 + 1/x)) = lim_(x->-oo) 1/(-sqrt (1 + 1/x)) = 1/(-1) = -1
Hence, evaluating the given limit yields lim_(x->-oo) x/(sqrt(x^2+x)) = -1.