Limits and Continuity: Analyzing Limits from Graphs

Q1. Use the graph of function f to find the following limits:

• limx→−3− f(x)

• limx→−3+ f(x)

• limx→−3 f(x)

• limx→0− f(x)

• limx→0+ f(x)

• limx→0 f(x)

• limx→3− f(x)

• limx→3+ f(x)

• limx→3 f(x)

• limx→−4− f(x)

• limx→−4+ f(x)

• limx→−4 f(x)

Background

Topic: Limits and Continuity

This question tests your ability to interpret limits from a graph, including one-sided limits and determining whether a limit exists at a given point. You will need to analyze the behavior of the function as x approaches specific values from the left and right.

Key Terms and Formulas:

• Limit: The value that a function approaches as the input approaches a certain point.

• One-sided limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

• Existence of a limit: The limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal.

• Discontinuity: If the left and right limits are not equal, the function is discontinuous at that point.

Step-by-Step Guidance

1. Examine the graph at each specified x-value (e.g., $x = -3$, $x = 0$, $x = 3$, $x = -4$). Identify the behavior of the function as x approaches these points from the left and right.

2. For each one-sided limit, trace the curve as x approaches the point from the left ($x \to a^-$) and from the right ($x \to a^+$). Note the y-value the function approaches.

3. Compare the left and right limits at each point. If they are equal, the two-sided limit exists and equals that value. If not, the two-sided limit does not exist (d.n.e.).

4. Pay attention to open and closed circles on the graph, which indicate whether the function is defined at that point or if there is a jump/discontinuity.

Try solving on your own before revealing the answer!

Final Answer:

The limits at each point can be read directly from the graph by following the steps above. For example, at $x = -3$, both one-sided limits approach the same value, so the two-sided limit exists. At $x = 0$, the left and right limits are different, so the two-sided limit does not exist. Continue this process for each x-value listed.

Remember: The existence of a limit depends on the agreement of the left and right limits, not just the function value at the point.