Calculus of a Single Variable, Chapter 3, 3.5, Section 3.5, Problem 20

You need to evaluate the limit, hence, you need to replace oo for x in equation:
lim_(x->-oo) (5/x - x/3) = 5/(-oo) - (-oo)/3 = 0 + oo
Since the result is indeterminate, you need bring the fractions to a common denominator:
lim_(x->-oo)(15 - x^2)/(3x)
You need to factor out x^2 to numerator:
lim_(x->-oo) (x^2(15/(x^2) - 1))/(3x)
Since lim_(x->-oo) 15/(x^2) = 0 , yields:
lim_(x->-oo) (x^(2-1))*(-1/3)= -1/3*lim_(x->-oo) (x^1) = -1/3*(-oo) = oo
Hence, evaluating the given limit yields lim_(x->-oo) (5/x - x/3) = oo.