To evaluate the limits in problem 6 using the given graph, we will analyze the behavior of the function $f(x)$ at the specified points. Here's the detailed analysis based on the visible portions of the graph:

Evaluating the Limits

a) $\lim_{x \to -\infty} f(x)$:

• As $x$ approaches $-\infty$, the function appears to approach a horizontal asymptote. From the graph, $f(x)$ seems to approach $y = -2$.
• Answer: $\lim_{x \to -\infty} f(x) = -2$

b) $\lim_{x \to 0^-} f(x)$:

• As $x$ approaches 0 from the left, $f(x)$ appears to approach 0.
• Answer: $\lim_{x \to 0^-} f(x) = 0$

c) $\lim_{x \to 0^+} f(x)$:

• As $x$ approaches 0 from the right, $f(x)$ also appears to approach 0.
• Answer: $\lim_{x \to 0^+} f(x) = 0$

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

• As $x$ approaches 1 from the left, the function value approaches approximately 2.
• Answer: $\lim_{x \to 1^-} f(x) = 2$

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

• As $x$ approaches 1 from the right, the function value also approaches approximately 2.
• Answer: $\lim_{x \to 1^+} f(x) = 2$

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

• As $x$ approaches 3, the function value approaches 0.
• Answer: $\lim_{x \to 3} f(x) = 0$

g) $\lim_{x \to 4} f(x)$:

• As $x$ approaches 4, the function value appears to be around -1.
• Answer: $\lim_{x \to 4} f(x) = -1$

h) $\lim_{x \to +\infty} f(x)$:

• As $x$ approaches $+\infty$, the function appears to approach a horizontal asymptote. From the graph, $f(x)$ seems to approach $y = 2$.
• Answer: $\lim_{x \to +\infty} f(x) = 2$

Points of Discontinuity

To determine the points of discontinuity, we look for points where the function does not meet the criteria for continuity:

1. Check for Jump Discontinuities: Evaluate the function's behavior around specific $x$-values where the limits from the left and right do not match.
2. Check for Infinite Discontinuities: Evaluate points where the function approaches infinity.
3. Check for Removable Discontinuities: Evaluate points where the function is not defined but could be made continuous by appropriately defining $f(x)$.

From the given graph and limits:

• No obvious discontinuities are visible within the inspected range, as all calculated limits at the specified points seem to be consistent with the function's behavior.

Would you like any more details or explanations on this problem?

Here are some questions you might want to ask next:

1. How do you handle piecewise functions when evaluating limits?
2. What are the different types of discontinuities in a function?
3. Can you explain how to determine vertical and horizontal asymptotes from a graph?
4. How do you use the definition of continuity to prove a function is continuous?
5. Can you provide examples of real-world applications of continuity and limits?

Tip:

When evaluating limits from a graph, carefully observe the behavior of the function as it approaches the specified points from both the left and right sides.