Calculus of a Single Variable, Chapter 3, 3.5, Section 3.5, Problem 32

You need to evaluate the limit, hence, you need to replace oo for x in equation:
lim_(x->-oo) (sqrt(x^4-1))/(x^3-1) = (sqrt(oo^4-1))/((-oo)^3-1)= (oo)/(-oo)
Since the result is indeterminate, you need to factor out x^4 at numerator and x^3 at denominator:
lim_(x->-oo) (sqrt (x^4(1 - 1/x^4)))/(x^3(1-1/x^3)) =
lim_(x->-oo)(x^2sqrt (x^4(1 - 1/x^4)))/(x^3(1-1/x^3)) =
Since lim_(x->-oo) 1/(x^3) = 0 and lim_(x->oo) 1/x^4 = 0 , yields:
lim_(x->oo) (x^(2-3))*(1/1)= 1*lim_(x->oo) 1/x=1*1/(oo) =1*0=0
Hence, evaluating the given limit yields lim_(x->-oo) (sqrt(x^4-1))/(x^3-1) = 0.