Answer
a) $3.$
b) $2.$
c) $2.$
Work Step by Step
a)By Theorem $1.3: $ $\lim\limits_{x\to c} p(x)=p(c)$
$\lim\limits_{x\to1}(4-x^2)=4-1^2=3.$
b)Using Theorem $1.4: \lim\limits_{x\to c}\sqrt[n]{x}=\sqrt[n]{c}.$
and Theorem $1.5: \lim\limits_{x\to c}(f(g(x)))=f(\lim\limits_{x\to c}g(x)).$
$\lim\limits_{x\to3}\sqrt{x+1}=\sqrt{\lim\limits_{x\to3}(x+1)}=\sqrt{3+1}=2.$
c)By Theorem $1.5: \lim\limits_{x\to c}(f(g(x)))=f(\lim\limits_{x\to c}g(x)).$
$\lim\limits_{x\to1}g(f(x))=g(\lim\limits_{x\to1}f(x))=g(3)=2.$
