Answer
the graphs of f and g are given, use them to evaluate each limit if it exists. If the limit does not exist, explain why.
(a) 1
(b) the limit of the difference does not exist because g(x) doesn't have limit
(c) 2
(d) The limit of the quotient doesn't exist because the denominator is 0
(e) -4
(f) 5
Work Step by Step
(a) $\lim \limits_{x \to 2}[f(x) + g(x)] $
Apply the sum law
$$\lim \limits_{x \to 2}f(x) + \lim \limits_{x \to 2} g(x)= -1 + 2 = 1 $$
(b) $\lim\limits_{x \to 0} [f(x) - g(x)]$
Apply the difference law
$$\lim\limits_{x \to 0} f(x) - \lim\limits_{x \to 0} g(x) = 2 - \nexists$$
Because the limit of $g(x)$ (given the graph) doesn't exist, the difference with $f(x)$ and $g(x)$ don't exist
The limit doesn't exist
(c) $\lim\limits_{x \to -1} [f(x) *g(x)]$
Apply the product law
$$\lim\limits_{x \to -1} f(x) *\lim\limits_{x \to -1}g(x) = 1 * 2=2$$
(d) $\lim\limits_{x \to 3} [\frac{f(x)}{g(x)}]$
Apply the quotient law
$$\frac{\lim\limits_{x \to 3} f(x)}{\lim\limits_{x \to 3} g(x)}= \frac{1}{0} = \nexists$$ the limit doesn't exist because $g(x)\neq 0$
(e) $\lim\limits_{x \to 2} [x^2*f(x)]$
Apply the product law
Note: $x^2 $is just other function
$$\lim\limits_{x \to 2} x^2*\lim\limits_{x \to 2} f(x) = (2)^2 * (-1) = 4*(-1) = -4$$
(f) $f(-1) + \lim\limits_{x \to -1} g(x) $
Apply the sum law
Note: the function at $x=-1$ is defined at $3$. We can substitute
$$3 + 2 =5$$
