the graph of a function is given. use the graph to find the indicated limits and function values, or state…
the graph of a function is given. use the graph to find the indicated limits and function values, or state that the limit or function value does not exist. a. lim f(x) x→−3− b. lim f(x) x→−3+ c. lim f(x) x→−3 d. f(−3) e. lim f(x) x→−1− f. lim f(x) x→−1+ g. lim f(x) x→−1 h. f(−1) i. lim f(x) x→3− j. lim f(x) x→3+ k. lim f(x) x→3 l. f(3)
Answer
Explanation:
Step1: Recall limit - from - the - left concept
To find $\lim_{x\rightarrow a^{-}}f(x)$, we look at the values of the function as $x$ approaches $a$ from the left - hand side of the graph.
Step2: Recall limit - from - the - right concept
To find $\lim_{x\rightarrow a^{+}}f(x)$, we look at the values of the function as $x$ approaches $a$ from the right - hand side of the graph.
Step3: Recall two - sided limit concept
$\lim_{x\rightarrow a}f(x)$ exists if and only if $\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)$. To find $f(a)$, we look at the $y$ - value of the point on the graph with $x = a$.
a. $\lim_{x\rightarrow - 3^{-}}f(x)$
As $x$ approaches $-3$ from the left, the $y$ - values approach $2$. So, $\lim_{x\rightarrow - 3^{-}}f(x)=2$.
b. $\lim_{x\rightarrow - 3^{+}}f(x)$
As $x$ approaches $-3$ from the right, the $y$ - values approach $2$. So, $\lim_{x\rightarrow - 3^{+}}f(x)=2$.
c. $\lim_{x\rightarrow - 3}f(x)$
Since $\lim_{x\rightarrow - 3^{-}}f(x)=\lim_{x\rightarrow - 3^{+}}f(x) = 2$, then $\lim_{x\rightarrow - 3}f(x)=2$.
d. $f(-3)$
The point on the graph with $x=-3$ has $y = 2$. So, $f(-3)=2$.
e. $\lim_{x\rightarrow - 1^{-}}f(x)$
As $x$ approaches $-1$ from the left, the $y$ - values approach $3$. So, $\lim_{x\rightarrow - 1^{-}}f(x)=3$.
f. $\lim_{x\rightarrow - 1^{+}}f(x)$
As $x$ approaches $-1$ from the right, the $y$ - values approach $1$. So, $\lim_{x\rightarrow - 1^{+}}f(x)=1$.
g. $\lim_{x\rightarrow - 1}f(x)$
Since $\lim_{x\rightarrow - 1^{-}}f(x)\neq\lim_{x\rightarrow - 1^{+}}f(x)$, $\lim_{x\rightarrow - 1}f(x)$ does not exist.
h. $f(-1)$
The point on the graph with $x = - 1$ has $y=3$. So, $f(-1)=3$.
i. $\lim_{x\rightarrow3^{-}}f(x)$
As $x$ approaches $3$ from the left, the $y$ - values approach $4$. So, $\lim_{x\rightarrow3^{-}}f(x)=4$.
j. $\lim_{x\rightarrow3^{+}}f(x)$
As $x$ approaches $3$ from the right, the $y$ - values approach $4$. So, $\lim_{x\rightarrow3^{+}}f(x)=4$.
k. $\lim_{x\rightarrow3}f(x)$
Since $\lim_{x\rightarrow3^{-}}f(x)=\lim_{x\rightarrow3^{+}}f(x)=4$, then $\lim_{x\rightarrow3}f(x)=4$.
l. $f(3)$
The point on the graph with $x = 3$ has $y=4$. So, $f(3)=4$.
Answer:
a. $2$ b. $2$ c. $2$ d. $2$ e. $3$ f. $1$ g. Does not exist h. $3$ i. $4$ j. $4$ k. $4$ l. $4$