Question

1. $lim_{x

ightarrow - 1}f(x)=$ help (limits)

1. $lim_{x

ightarrow1^{-}}f(x)=$

1. $lim_{x

ightarrow1}f(x)=$

1. $lim_{x

ightarrow4}f(x)=$

1. $f(4)=$

Explanation:

Step1: Recall left - hand limit definition

The left - hand limit $\lim_{x
ightarrow a^{-}}f(x)$ is the value the function approaches as $x$ approaches $a$ from the left. For $\lim_{x
ightarrow1^{-}}f(x)$, we look at the values of $y$ as $x$ approaches $1$ from the left side of the graph. As $x$ approaches $1$ from the left, $y$ approaches $2$.

Step2: Recall right - hand limit definition

The right - hand limit $\lim_{x
ightarrow a^{+}}f(x)$ is the value the function approaches as $x$ approaches $a$ from the right. For $\lim_{x
ightarrow1^{+}}f(x)$, we look at the values of $y$ as $x$ approaches $1$ from the right side of the graph. As $x$ approaches $1$ from the right, $y$ approaches $- 1$.

Step3: Recall two - sided limit condition

The two - sided limit $\lim_{x
ightarrow a}f(x)$ exists if and only if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)$. Since $\lim_{x
ightarrow1^{-}}f(x) = 2$ and $\lim_{x
ightarrow1^{+}}f(x)=-1$, $\lim_{x
ightarrow1}f(x)$ does not exist.

Step4: Analyze limit as $x$ approaches $4$

For $\lim_{x
ightarrow4}f(x)$, as $x$ approaches $4$ from both the left and the right, the function values approach $3$. So $\lim_{x
ightarrow4}f(x)=3$.

Step5: Find the function value at $x = 4$

The solid dot at $x = 4$ has a $y$ - value of $1$. So $f(4)=1$.

Answer:

1. Does not exist
2. $2$
3. $-1$
4. $3$
5. $1$