How do you determine the limit of # (x^2+4)/( x^3-1)# as x approaches 1-?

1 Answer
Feb 25, 2017

The limit does not exist because as #x# approaches #1# from the left, the quotient decreases without bound. We write #lim_(xrarr1^-)(x^2+4)/(x^3-1) = -oo#

Explanation:

As #xrarr1#, the numerator goes to #5#, which is (of course) positive.

As #xrarr1^-#, we have #x < 1#, so #x^3 < 1# and #x^3-1# approaches #0# through negative values.

A quotient whose numerator is approaching a positive limit and whose denominator is going to #0# through negative fraction is DECREASING without bound.