Answer
(a) Sum Law: $\lim\limits_{x \to a}[f(x)+g(x)]=\lim\limits_{x \to a}f(x)+\lim\limits_{x \to a}g(x)$
(b) Difference Law: $\lim\limits_{x \to a}[f(x)-g(x)]=\lim\limits_{x \to a}f(x)-\lim\limits_{x \to a}g(x)$
(c) Constant Multiple Law: $\lim\limits_{x \to a}[cf(x)]=c\lim\limits_{x \to a}f(x)$
(d) Product Law: $\lim\limits_{x \to a}[f(x)g(x)]=\lim\limits_{x \to a}f(x)\times\lim\limits_{x \to a}g(x)$
(e) Quotient Law: $\lim\limits_{x \to a}\frac{f(x)}{g(x)}=\frac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)}$
Note, only if $\lim\limits_{x \to a}g(x)\ne0$
(f) Power Law: $\lim\limits_{x \to a}[f(x)]^n=[\lim\limits_{x \to a}f(x)]^n$
Note, $\underline n$ is a positive integer.
(g) Root Law: $\lim\limits_{x \to a}\sqrt[n] {f(x)}=\sqrt[n] {\lim\limits_{x \to a}f(x)}$
Note, $\underline n$ is a positive integer. If $n$ is even, we assume that $\lim\limits_{x \to a}f(x) > 0$
Work Step by Step
As stated in the chapter 2.3, we know the following:
Let's suppose that $c$ is a constant and the limits $\lim\limits_{x \to a}f(x)$ and $\lim\limits_{x \to a}g(x)$ exist.
(a) Sum Law: $\lim\limits_{x \to a}[f(x)+g(x)]=\lim\limits_{x \to a}f(x)+\lim\limits_{x \to a}g(x)$
The limit of sum is the sum of the limits.
(b) Difference Law: $\lim\limits_{x \to a}[f(x)-g(x)]=\lim\limits_{x \to a}f(x)-\lim\limits_{x \to a}g(x)$
The limit of a difference is the difference of the limits.
(c) Constant Multiple Law: $\lim\limits_{x \to a}[cf(x)]=c\lim\limits_{x \to a}f(x)$
The limit of a constant times a function is the constant times the limit of the function.
(d) Product Law: $\lim\limits_{x \to a}[f(x)g(x)]=\lim\limits_{x \to a}f(x)\times\lim\limits_{x \to a}g(x)$
The limit of a product is the product of the limits.
(e) Quotient Law: $\lim\limits_{x \to a}\frac{f(x)}{g(x)}=\frac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)}$
Note, only if $\lim\limits_{x \to a}g(x)\ne0$
The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).
(f) Power Law: $\lim\limits_{x \to a}[f(x)]^n=[\lim\limits_{x \to a}f(x)]^n$
Note, $\underline n$ is a positive integer.
(g) Root Law: $\lim\limits_{x \to a}\sqrt[n] {f(x)}=\sqrt[n] {\lim\limits_{x \to a}f(x)}$
Note, $\underline n$ is a positive integer. If $n$ is even, we assume that $\lim\limits_{x \to a}f(x) > 0$
