Answer
a) $3.$
b) $\frac{3}{2}.$
c) $729.$
d) $9.$
Work Step by Step
Using Theorem $1.2$ and Theorem $1.5: \lim\limits_{x\to c}(f(g(x)))=f(\lim\limits_{x\to c}g(x)):$
a) $\lim\limits_{x\to c}\sqrt[3]{f(x)}=\sqrt[3]{\lim\limits_{x\to c}f(x)}=\sqrt[3]{27}=3.$
b) $\lim\limits_{x\to c}\dfrac{f(x)}{18}=\dfrac{\lim\limits_{x\to c}f(x)}{\lim\limits_{x\to c}(18)}=\dfrac{27}{18}=\dfrac{3}{2}.$
c)$\lim\limits_{x\to c}[f(x)]^2=[\lim\limits_{x\to c}f(x)]^2=27^2=729.$
d) $\lim\limits_{x\to c}[f(x)]^{\frac{2}{3}}=[\lim\limits_{x\to c}f(x)]^{\frac{2}{3}}=27^{\frac{2}{3}}=9.$
