We have to find the value of lim x-->0 [(x - 64)^(1/3) - 4]/x

lim x-->0 [(x - 64)^(1/3) - 4]/x

If we substitute x = 0 in the denominator we get 0 and substituting x in the numerator gives -4 - 4 = -8.

As x tends to 0 the given expression tends to an exceedingly large negative value or to -infinity.

**The required value of lim x-->0 [(x - 64)^(1/3) - 4]/x = -inf.**

In other words, we'll have to evaluate the limit of the fraction, if x approaches to 0:

We'll replace x by 0, we'll get:

lim [(x - 64)^(1/3) - 4]/x = [(0 - 64)^(1/3) - 4]/0

lim [(x - 64)^(1/3) - 4]/x = [(-64)^(1/3) - 4]/0

But (-64)^(1/3) = -4

lim [(x - 64)^(1/3) - 4]/x = (-4 - 4)/0

lim [(x - 64)^(1/3) - 4]/x = -8/0

lim [(x - 64)^(1/3) - 4]/x = - infinite

**When x approches to 0, the value of the fraction [(x - 64)^(1/3) - 4]/x tends to - infinite.**