Please show all steps, thank you. selecting an appropriate strategy to evaluate a limit of the following.

Question

b) Evaluate lim x → 0 (x+8)^1/3- 2 / x ( (x+8)^1/3 - 2 all over x ) The numerator would be (x+8)^1/3 - 2 (the - 2 is not with the exponent) and the denominator is x.

Answer 1

Recall that f'(c) = lim_x→c [(f(x) - f(c)) / (x - c)]

Let f(x) = (x+8)^1/3 and c = 0. Then, f(c) = f(0) = 2 and f'(x) = (1/3)(x+8)^-2/3. So, f'(c) = f'(0) = 1/12.

Therefore, the value of the given limit is 1/12.