Let's work through the one-sided limits as requested based on the graph of $g(x)$ in Figure 1.14.

Problem 3: $\lim_{x \to 1^-} g(x)$

This is the limit of $g(x)$ as $x$ approaches 1 from the left (denoted by the minus sign).

• As we approach $x = 1$ from the left (i.e., values slightly less than 1), the graph shows that $g(x)$ is moving toward the point $(1, 1)$.
• Therefore, $\lim_{x \to 1^-} g(x) = 1$.

Problem 4: $\lim_{x \to 1^+} g(x)$

This is the limit of $g(x)$ as $x$ approaches 1 from the right (denoted by the plus sign).

• As $x$ approaches 1 from the right (i.e., values slightly greater than 1), the graph shows a jump discontinuity, but the function is approaching 0.
• Therefore, $\lim_{x \to 1^+} g(x) = 0$.

Problem 5: $\lim_{x \to 3^-} g(x)$

This is the limit of $g(x)$ as $x$ approaches 3 from the left.

• As we approach $x = 3$ from the left, the graph approaches the value of 1.
• Therefore, $\lim_{x \to 3^-} g(x) = 1$.

Problem 6: $\lim_{x \to 3^+} g(x)$

This is the limit of $g(x)$ as $x$ approaches 3 from the right.

• As $x$ approaches 3 from the right, the graph shows that $g(x)$ jumps up and tends to 2.
• Therefore, $\lim_{x \to 3^+} g(x) = 2$.

Would you like further clarification or more details about these problems?

Here are some related questions:

1. What is the general concept of one-sided limits?
2. What does a jump discontinuity mean in terms of limits?
3. How do one-sided limits help in understanding continuity at a point?
4. How would the limits change if the graph had no discontinuities?
5. What happens to a function’s overall limit if the one-sided limits don’t match?

Tip: To check if a function is continuous at a point, ensure that both the left-hand and right-hand limits exist and are equal, and that they match the function's value at that point.