Evaluate the following: limit lim as h approaches 0 of {[(8+h)^(1/3) -2] /h}

lim as h approaches 0 of {[(8+h)^(1/3) -2] /h}

Recall that f '(a) = lim(h→0) [f(a+h) - f(a)]/h.
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In this case, observe that
lim(h→0) [(8+h)^(1/3) - 2] / h = lim(h→0) [(8+h)^(1/3) - 8^(1/3)] / h.

So, in this case, we have a = 8 and f(x) = x^(1/3).

By using the definition of the derivative, this limit equals
(d/dx) x^(1/3) {at x = 8} = (1/3)x^(-2/3) {at x = 8} = 1/12.

I hope this helps!