Answer
a) $\lim\limits_{x\to 2}[f(x)+g(x)]=1$
b) $\lim\limits_{x\to 0}[f(x)-g(x)]$ does not exist.
c) $\lim\limits_{x\to -1}[f(x)g(x)]=2$
d) $\lim\limits_{x\to 3}\frac{f(x)}{g(x)}$ does not exist.
e) $\lim\limits_{x\to 2}[x^2f(x)]=-4$
f) $f(-1)+\lim\limits_{x\to -1}g(x)=5$
Work Step by Step
a) $\lim\limits_{x\to 2}f(x)=-1$ and $\lim\limits_{x\to 2}g(x)=2$
So,
$\lim\limits_{x\to 2}[f(x)+g(x)]=\lim\limits_{x\to 2}f(x)+\lim\limits_{x\to 2}g(x)=-1+2=1$
b) $\lim\limits_{x\to 0}f(x)=2$
But $\lim\limits_{x\to 0}g(x)$ does not exist since $\lim\limits_{x\to 0^+}g(x)=1$ while $\lim\limits_{x\to 0^-}g(x)=3$
Therefore, we calculate the one-side limit:
$\lim\limits_{x\to 0^+}[f(x)-g(x)]=2-1=1$
$\lim\limits_{x\to 0^-}[f(x)-g(x)]=2-3=-1$
So, $\lim\limits_{x\to 0}[f(x)-g(x)]$ does not exist.
c) $\lim\limits_{x\to -1}f(x)=1$ and $\lim\limits_{x\to -1}g(x)=2$
So,
$\lim\limits_{x\to -1}[f(x)g(x)]=\lim\limits_{x\to -1}f(x)\times\lim\limits_{x\to -1}g(x)=1\times2=2$
d) $\lim\limits_{x\to 3}f(x)=1$ and $\lim\limits_{x\to 3}g(x)=0$
$\lim\limits_{x\to 3}[\frac{f(x)}{g(x)}]=\frac{\lim\limits_{x\to 3}f(x)}{\lim\limits_{x\to 3}g(x)}$
However, $\lim\limits_{x\to 3}g(x)=0$
Therefore, $\lim\limits_{x\to 3}\frac{f(x)}{g(x)}$ is undefined and does not exist.
e) $\lim\limits_{x\to 2}f(x)=-1$ and $\lim\limits_{x\to 2}x^2=2^2=4$
$\lim\limits_{x\to 2}[x^2f(x)]=\lim\limits_{x\to 2}x^2\times\lim\limits_{x\to 2}f(x)=(-1)\times4=-4$
f) $f(-1)=3$ and $\lim\limits_{x\to -1}g(x)=2$
$f(-1)+\lim\limits_{x\to -1}g(x)=3+2=5$
