Analyzing Limits and Continuity from a Graph

Q5. For the function $g(x)$ shown in the graph, answer each part for each of the following values of $c$: $c = -2, -1, 0, 1, 2$:

• (a) $\lim_{x \to c^+} g(x)$

• (b) $\lim_{x \to c^-} g(x)$

• (c) $\lim_{x \to c} g(x)$

• (d) $g(c)$

• (e) Is $x = c$ in the domain of $g(x)$? Why or why not?

• (f) Is $g$ continuous at $x = c$? Why or why not?

Background

Topic: Limits and Continuity from a Graph

This question tests your ability to interpret a graph to determine one-sided and two-sided limits, function values, domain, and continuity at specific points.

Key Terms and Formulas

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

• Two-sided limit: $\lim_{x \to c} f(x)$ exists if and only if both one-sided limits exist and are equal

• Function value: $f(c)$ is the value of the function at $x = c$ (if defined)

• Continuity at $x = c$: $f$ is continuous at $c$ if $f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$

Step-by-Step Guidance

1. For each value of $c$ ($-2, -1, 0, 1, 2$), examine the graph of $g(x)$ at and around $x = c$.

2. To find $\lim_{x \to c^-} g(x)$, look at the $y$-values as $x$ approaches $c$ from the left. For $\lim_{x \to c^+} g(x)$, look as $x$ approaches $c$ from the right.

3. Determine if the two one-sided limits are equal. If so, the two-sided limit $\lim_{x \to c} g(x)$ exists and equals that value. If not, the two-sided limit does not exist.

4. Check if there is a filled (solid) dot at $x = c$ on the graph. If so, $g(c)$ is defined and equals the $y$-value of that dot. If there is an open circle, $g(c)$ is not defined at that $y$-value.

5. Decide if $x = c$ is in the domain of $g(x)$ by checking if $g(c)$ is defined (solid dot at $x = c$).

6. To determine continuity at $x = c$, check if $g(c)$ is defined, the two-sided limit exists, and $g(c)$ equals the limit. If any of these fail, $g$ is not continuous at $x = c$.

7. Repeat this process for each value of $c$.

Try solving on your own before revealing the answer!

Final Answer:

For each $c$ value, the limits, function values, domain status, and continuity are as follows (read from the graph):

• At $c = -2$: $\lim_{x \to -2^-} g(x) = 1$, $\lim_{x \to -2^+} g(x) = -2$, $\lim_{x \to -2} g(x)$ does not exist (one-sided limits not equal), $g(-2)$ is not defined (open circle), $x = -2$ is not in the domain, $g$ is not continuous at $x = -2$.

• At $c = -1$: $\lim_{x \to -1^-} g(x) = -2$, $\lim_{x \to -1^+} g(x) = 0$, $\lim_{x \to -1} g(x)$ does not exist, $g(-1) = 0$ (solid dot), $x = -1$ is in the domain, $g$ is not continuous at $x = -1$.

• At $c = 0$: $\lim_{x \to 0^-} g(x) = 0$, $\lim_{x \to 0^+} g(x) = 2$, $\lim_{x \to 0} g(x)$ does not exist, $g(0)$ is not defined (open circle), $x = 0$ is not in the domain, $g$ is not continuous at $x = 0$.

• At $c = 1$: $\lim_{x \to 1^-} g(x) = 2$, $\lim_{x \to 1^+} g(x) = -1$, $\lim_{x \to 1} g(x)$ does not exist, $g(1) = 2$ (solid dot), $x = 1$ is in the domain, $g$ is not continuous at $x = 1$.

• At $c = 2$: $\lim_{x \to 2^-} g(x) = -1$, $\lim_{x \to 2^+} g(x) = 1$, $\lim_{x \to 2} g(x)$ does not exist, $g(2) = 1$ (solid dot), $x = 2$ is in the domain, $g$ is not continuous at $x = 2$.

For each $c$, the function is not continuous because either the limit does not exist or the function value does not match the limit.