e19e08…: “Regardless of a spring house”

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1 : The entire graph of a function $f$ is given below.

The domain of $f$ is $\left [\answer {-4},\answer {7}\right ]$ .

The range of $f$ (from bottom to top) is:

$\left [\answer {-4},\answer {-2}\right ] \cup \left \{\answer {-1}\right \} \cup \left (\answer {0},\answer {2}\right ) \cup \left (\answer {2},\answer {6}\right ]$ .

Find the following limits, if they exist. If a limit does not exist, explain why.

1.1 : The limit as $x$ approaches $0$

The limit does exist. The limit does not exist because $f(0)$ does not exist. The limit does not exist because $f(0)$ is not close to the values of $f$ near 0. The limit does not exist because the left limit does not approach the same value as the right limit.

1.1.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {2}$

1.2 : The limit as $x$ approaches $3$ from the left.

The limit does exist. The limit does not exist because $f(3)$ does not exist. The limit does not exist because $f(3)$ is not close to the values of $f$ near 3. The limit does not exist because the left limit does not approach the same value as the right limit.

1.2.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {0}$

1.3 : The limit as $x$ approaches $3$ from the right.

1.3.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {-2}$

1.4 : The limit as $x$ approaches $3$ .

1.4.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer [format=string]{DNE}$

2 :

The entire graph of a function $f$ is given below.

Find the following values.

$\begin{align*} f(0) &= \answer{1}\\ f(2) &= \answer{3}\\ f(f(0)) &= \answer{3}\\ f(-1) &= \answer{1} \end{align*}$

Find the following limits, if they exist. If a limit does not exist, explain why.

2.1 : The limit as $x$ approaches $-2$

The limit does exist. The limit does not exist because $f$ is constant near -2. The limit does not exist because $f(-2)$ does not exist. The limit does not exist because the left limit does not approach the same value as the right limit.

2.1.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {1}$

2.2 : The limit as $x$ approaches $3$ from the right

The limit does exist. The limit does not exist because $f$ is constant near -3. The limit does not exist because $f$ is not defined to the left of -3. The limit does not exist because the left limit does not approach the same value as the right limit.

2.2.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {1}$

2.3 : The limit as $x$ approaches $2$

The limit does exist. The limit does not exist because the value of $f$ at 2 is very different from the value of $f$ near 2. The limit does not exist because $f(2)$ does not exist. The limit does not exist because the left limit does not approach the same value as the right limit.

2.3.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {2}$

2.4 : The limit as $x$ approaches $1$

The limit does exist. The limit does not exist because $f$ is not constant near 1. The limit does not exist because $f(1)$ does not exist. The limit does not exist because the left limit does not approach the same value as the right limit.

2.4.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {3}$

2.5 : The limit as $x$ approaches $0$ from the left

The limit does exist. The limit does not exist because $f$ is not defined to the right of 0. The limit does not exist because $f(0)$ does not exist. The limit does not exist because the left limit does not approach the same value as the right limit.

2.5.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {1}$

2.6 : The limit as $x$ approaches $0$ from the right

The limit does exist. The limit does not exist because $f$ is not defined to the left of 0. The limit does not exist because $f(0)$ does not exist. The limit does not exist because the left limit does not approach the same value as the right limit.

2.6.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer {4}$

2.7 : The limit as $x$ approaches $0$

The limit does exist. The limit does not exist because $f$ is not defined the same way on either side of 0. The limit does not exist because $f(0)$ does not exist. The limit does not exist because the left limit does not approach the same value as the right limit.

2.7.1 : $\text {The Value of the limit is: (Enter DNE if the limit does not exist) } \answer [format=string]{DNE}$