Answer
$6$
Work Step by Step
Consider $f(x)=\int_{-1}^{1} \dfrac{1}{x^{2/3}} dx$
or, $f(x)=\int_{-1}^{0} \dfrac{1}{x^{2/3}} dx +\int_{0}^{1} \dfrac{1}{x^{2/3}} dx$
Thus, $\lim\limits_{a \to 0^{-}} f(x)= \lim\limits_{a \to 0^{-}} \int_{-1}^{a} \dfrac{1}{x^{2/3}} dx+ \lim\limits_{a \to 0^{+}} \int_{0}^{1} \dfrac{1}{x^{2/3}} dx=\lim\limits_{a \to 0^{-}} [3a^{1/3} -(-3)]+\lim\limits_{a \to 0^{+}} [3-3a^{1/3} ]$
or, $\lim\limits_{a \to 0^{-}} [3a^{1/3} +3]+\lim\limits_{a \to 0^{+}} [3-3a^{1/3} = 3+3=6$
