Let’s walk through the graph to evaluate the limits based on the function $y = f(x)$.

The task asks for:

1. $\lim_{x \to 1} f(x)$
2. $\lim_{x \to 0} f(x)$
3. $\lim_{x \to 4^-} f(x)$

Step-by-Step Evaluation

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

   • As $x$ approaches 1 from both sides, the function values appear to get closer to 5 (where the curve meets the point at $(1, 5)$).
   • Thus, $\lim_{x \to 1} f(x) = 5$.

2. Evaluating $\lim_{x \to 0} f(x)$:

   • As $x$ approaches 0 from both the left and right, the function value stabilizes around 3 (since the graph passes smoothly through $(0, 3)$).
   • So, $\lim_{x \to 0} f(x) = 3$.

3. Evaluating $\lim_{x \to 4^-} f(x)$:

   • When $x$ approaches 4 from the left side, the function value seems to approach -3.
   • There is no discontinuity as we approach 4 from the left.
   • Therefore, $\lim_{x \to 4^-} f(x) = -3$.

Answer Matching

Based on our evaluations:

• $\lim_{x \to 1} f(x) = 5$
• $\lim_{x \to 0} f(x) = 3$
• $\lim_{x \to 4^-} f(x) = -3$

Thus, the correct answer is the option:

• $\lim_{x \to 1} f(x) = 5$, $\lim_{x \to 0} f(x) = 3$, $\lim_{x \to 4^-} f(x) = -3$

Follow-Up Questions:

1. What is the definition of a left-hand and right-hand limit?
2. How can you identify discontinuities in a graph based on the limit behavior?
3. What does it mean if a limit does not exist at a point?
4. How do piecewise functions affect limit evaluations?
5. What role do removable discontinuities play in limits?

Tip: Always compare the left-hand and right-hand limits at a point. If they differ, the limit at that point does not exist!