Question

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

Answer 1

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.