Chapter 2.3, Problem 75E

Useful factorization formula Calculate the following limits using the factorization formula

$x^n-a^n=(x-a)(x^{n-1}+x^{n-2}a+x^{n-3}{a}^2+\cdots +x{a}^{n-2}+a^{n-1}),$

where n is a positive integer and a is a real number.

75. $\underset{x\to 1}{\lim }\frac{\sqrt[3]{x}-1}{x-1}$(Hint: $x-1={(\sqrt[3]{x})}^3-{(1)}^3$.)