+42 \(\lim_{x\rightarrow 1}(\sqrt[3]{x}-1)/(x-1)\)
Wolfram alpha did not explain this to me so well. How the heck do we end up with \(1/(1+\sqrt[3]{x}+{x}^{2/3})\)??
Please show me in great detail how to crack down something this complicated.
Thanks!
Find the following limit:
lim_(x->1) (x^(1/3) - 1)/(x - 1)
Simplify the expression inside the limit.
(x^(1/3) - 1)/(x - 1) = (x^(1/3) - 1)/(x - 1):
lim_(x->1) (x^(1/3) - 1)/(x - 1)
Rationalize the expression.
(x^(1/3) - 1)/(x - 1) = ((x^(1/3) - 1) (1 + x^(1/3) + x^(2/3)))/((x - 1) (1 + x^(1/3) + x^(2/3))) = 1/(1 + x^(1/3) + x^(2/3)):
lim_(x->1) 1/(1 + x^(1/3) + x^(2/3))
The limit of a continuous function at a point is just its value there.
lim_(x->1) 1/(1 + x^(1/3) + x^(2/3)) = 1/(1 + 1^(1/3) + 1^(2/3)) = 1/3:
=1/3
