Question

assignment 2: problem 7 (1 point)
let f be the function whose graph is shown below.
evaluate each of the following expressions.
(if a limit does not exist or is undefined, enter \dne\.)

1. $lim_{x

ightarrow - 1^{-}}f(x)=$

1. $lim_{x

ightarrow - 1^{+}}f(x)=$

1. $lim_{x

ightarrow - 1}f(x)=$

1. $f(-1)=$
2. $lim_{x

ightarrow 1^{-}}f(x)=$

1. $lim_{x

ightarrow 1^{+}}f(x)=$

1. $lim_{x

ightarrow 1}f(x)=$

1. $lim_{x

ightarrow 3}f(x)=$

1. $f(3)=$

the graph of $y = f(x)$.

Explanation:

Step1: Analyze left - hand limit at $x=-1$

As $x\to - 1^{-}$, follow the graph from the left, $y = 3$.

Step2: Analyze right - hand limit at $x=-1$

As $x\to - 1^{+}$, follow the graph from the right, $y=-3$.

Step3: Analyze limit at $x = - 1$

Since left - hand and right - hand limits are different, $\lim_{x\to - 1}F(x)$ DNE.

Step4: Find $F(-1)$

There is a closed - dot at $(-1,3)$, so $F(-1)=3$.

Step5: Analyze left - hand limit at $x = 1$

As $x\to1^{-}$, $y = 1$.

Step6: Analyze right - hand limit at $x = 1$

As $x\to1^{+}$, $y = 1$.

Step7: Analyze limit at $x = 1$

Since left - hand and right - hand limits are equal, $\lim_{x\to1}F(x)=1$.

Step8: Analyze limit at $x = 3$

As $x\to3$, $y=-1$.

Step9: Find $F(3)$

There is an open - dot at $x = 3$, so $F(3)$ DNE.

Answer:

1. $3$
2. $-3$
3. DNE
4. $3$
5. $1$
6. $1$
7. $1$
8. $-1$
9. DNE