Let's break down the graph and solve each part of the question step by step.

Observing the Graph at $x = 1$:

• The function seems to have a discontinuity at $x = 1$.
• There's a jump discontinuity, which means the left-hand limit ($\lim_{x \to 1^-} f(x)$) and the right-hand limit ($\lim_{x \to 1^+} f(x)$) are different.

Now, let's compute each value:

1. $\lim_{x \to 1^-} f(x)$:

This is the limit as $x$ approaches 1 from the left. Looking at the graph, as $x$ approaches 1 from the left side (from values like $x = 0.5, 0.9$), the function tends to a value of 3.

Thus, $\lim_{x \to 1^-} f(x) = 3$

2. $\lim_{x \to 1^+} f(x)$:

This is the limit as $x$ approaches 1 from the right. From the graph, as $x$ approaches 1 from the right side (from values like $x = 1.1, 1.5$), the function tends to a value of -4.

Thus, $\lim_{x \to 1^+} f(x) = -4$

3. $\lim_{x \to 1} f(x)$:

The overall limit at $x = 1$ exists only if both the left-hand and right-hand limits are equal. Since $\lim_{x \to 1^-} f(x) = 3$ and $\lim_{x \to 1^+} f(x) = -4$, these are not equal, so the limit does not exist.

Thus, $\lim_{x \to 1} f(x) = \text{DNE (Does Not Exist)}$

4. $f(1)$:

The value of the function at $x = 1$ is given by the point at $x = 1$. From the graph, there is a solid dot at $x = 1$ corresponding to the value -2.

Thus, $f(1) = -2$

Summary of Answers:

1. $\lim_{x \to 1^-} f(x) = 3$
2. $\lim_{x \to 1^+} f(x) = -4$
3. $\lim_{x \to 1} f(x) = \text{DNE}$
4. $f(1) = -2$

Let me know if you want more details or have further questions!

Here are 5 related questions:

1. What is the difference between jump and removable discontinuities?
2. How would the graph change if $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x)$?
3. Can a function have a value at $x = 1$ even if $\lim_{x \to 1} f(x)$ does not exist?
4. How do you calculate the limit at a point of continuity versus discontinuity?
5. How do you graphically identify the left-hand and right-hand limits?

Tip: When analyzing a discontinuous function, always check the left-hand and right-hand limits separately to identify the type of discontinuity!