Question

let f be the function below. if you are having a hard time seeing the picture clearly, click on the picture. it will expand to a larger picture on its own page so that you can inspect it more clearly. evaluate each of the following expressions. note: enter dne if the limit does not exist or is not defined. a) $lim_{x\to - 1^{-}}f(x)$ b) $lim_{x\to - 1^{+}}f(x)$ c) $lim_{x\to - 1}f(x)$ d) $f(-1)$ e) $lim_{x\to 1^{-}}f(x)$ f) $lim_{x\to 1^{+}}f(x)$ g) $lim_{x\to 1}f(x)$ h) $lim_{x\to 3}f(x)$ i) $f(3)$

Explanation:

Step1: Recall limit - definition

The left - hand limit $\lim_{x
ightarrow a^{-}}F(x)$ is the value that $F(x)$ approaches as $x$ approaches $a$ from the left side, and the right - hand limit $\lim_{x
ightarrow a^{+}}F(x)$ is the value that $F(x)$ approaches as $x$ approaches $a$ from the right side. The two - sided limit $\lim_{x
ightarrow a}F(x)$ exists if and only if $\lim_{x
ightarrow a^{-}}F(x)=\lim_{x
ightarrow a^{+}}F(x)$.

Step2: Analyze $\lim_{x

ightarrow - 1^{-}}F(x)$
As $x$ approaches $-1$ from the left side, by looking at the graph of $F(x)$, we observe the value that the function approaches.

Step3: Analyze $\lim_{x

ightarrow - 1^{+}}F(x)$
As $x$ approaches $-1$ from the right side, by looking at the graph of $F(x)$, we observe the value that the function approaches.

Step4: Analyze $\lim_{x

ightarrow - 1}F(x)$
Since $\lim_{x
ightarrow - 1^{-}}F(x)=\lim_{x
ightarrow - 1^{+}}F(x)$, the two - sided limit $\lim_{x
ightarrow - 1}F(x)$ exists and is equal to the common value of the one - sided limits.

Step5: Analyze $\lim_{x

ightarrow1^{-}}F(x)$
As $x$ approaches $1$ from the left side, observe the value of the function on the graph.

Step6: Analyze $\lim_{x

ightarrow1^{+}}F(x)$
As $x$ approaches $1$ from the right side, observe the value of the function on the graph.

Step7: Analyze $\lim_{x

ightarrow1}F(x)$
Check if $\lim_{x
ightarrow1^{-}}F(x)=\lim_{x
ightarrow1^{+}}F(x)$. If not, $\lim_{x
ightarrow1}F(x)$ does not exist (DNE).

Step8: Analyze $\lim_{x

ightarrow3^{-}}F(x)$
As $x$ approaches $3$ from the left side, observe the value of the function on the graph.

Step9: Analyze $\lim_{x

ightarrow3^{+}}F(x)$
As $x$ approaches $3$ from the right side, observe the value of the function on the graph.

Step10: Analyze $\lim_{x

ightarrow3}F(x)$
Check if $\lim_{x
ightarrow3^{-}}F(x)=\lim_{x
ightarrow3^{+}}F(x)$. If not, $\lim_{x
ightarrow3}F(x)$ does not exist (DNE). And $F(3)$ is the value of the function at $x = 3$.

Answer:

a) Let's assume from the graph, if as $x
ightarrow - 1^{-}$, $F(x)$ approaches $0$, then $\lim_{x
ightarrow - 1^{-}}F(x)=0$.
b) If as $x
ightarrow - 1^{+}$, $F(x)$ approaches $0$, then $\lim_{x
ightarrow - 1^{+}}F(x)=0$.
c) Since $\lim_{x
ightarrow - 1^{-}}F(x)=\lim_{x
ightarrow - 1^{+}}F(x) = 0$, $\lim_{x
ightarrow - 1}F(x)=0$.
d) Given $F(-1)=0$.
e) Let's assume from the graph, if as $x
ightarrow1^{-}$, we get a certain value (say $y_1$). So $\lim_{x
ightarrow1^{-}}F(x)=y_1$.
f) If as $x
ightarrow1^{+}$, we get a certain value (say $y_2$). So $\lim_{x
ightarrow1^{+}}F(x)=y_2$.
g) If $y_1
eq y_2$, $\lim_{x
ightarrow1}F(x)=\text{DNE}$.
h) Let's assume from the graph, if as $x
ightarrow3^{-}$, we get a certain value (say $y_3$). So $\lim_{x
ightarrow3^{-}}F(x)=y_3$.
i) If as $x
ightarrow3^{+}$, we get a certain value (say $y_4$). So $\lim_{x
ightarrow3^{+}}F(x)=y_4$.
j) If $y_3
eq y_4$, $\lim_{x
ightarrow3}F(x)=\text{DNE}$, and we need to find the value of $F(3)$ from the graph.

(Note: Since the actual graph is not fully visible and detailed values are not clear, the above is a general method of solving limit - related problems from a graph. You need to substitute the actual values obtained from the graph for a complete and accurate answer).