Let's analyze the graph step by step to answer the questions given.

Analyzing the Graph

The graph represents the function $f(x)$ with several key points where it has specific behaviors that we will evaluate.

a) Calculating the left and right limits at the given points:

1. $\lim_{x \to 0^-} f(x)$: Observing the graph as $x$ approaches 0 from the left, $f(x)$ approaches 2.

2. $\lim_{x \to 0^+} f(x)$: As $x \to 0^+$, $f(x)$ still approaches 2. Therefore, $\lim_{x \to 0} f(x) = 2.$

3. $\lim_{x \to 18^-} f(x)$: As $x \to 18$ from the left, $f(x)$ approaches the value $6$.

4. $\lim_{x \to 18^+} f(x)$: From the right side of $x = 18$, the function approaches the value $4$.

   Therefore, since the left-hand and right-hand limits do not match at $x = 18$, the overall limit does not exist at this point.

Summary of limits:

• $\lim_{x \to 0} f(x) = 2$
• $\lim_{x \to 18^-} f(x) = 6$
• $\lim_{x \to 18^+} f(x) = 4$

b) At which of the points $x = 0, 16, 18, 19$ does the limit of $f(x)$ exist?

From the observations:

• At $x = 0$, the left-hand and right-hand limits are equal: exists.
• At $x = 16$, the function does not show any jump or discontinuity, so the limit exists.
• At $x = 18$, the left and right-hand limits differ: does not exist.
• At $x = 19$, the function appears to have a definite value from both sides.

Thus, the limit exists at: $x = 0, 16, 19$.

c) At which of the points $x = 0, 16, 18, 19$ is the function continuous?

A function is continuous at a point $x = a$ if:

1. The limit $\lim_{x \to a} f(x)$ exists.
2. $f(a)$ exists.
3. The limit equals $f(a)$.

• At $x = 0$, $f(0) = 2$, and the limit $\lim_{x \to 0} f(x) = 2$. Hence, $f(x)$ is continuous at $x = 0$.
• At $x = 16$, there is no jump or break, and $f(x)$ appears to be continuous.
• At $x = 18$, the function has a jump discontinuity since left and right-hand limits differ. So it is not continuous.
• At $x = 19$, although there is a closed dot, the function is continuous as both left-hand and right-hand limits match the function value at $x = 19$.

Therefore, the function is continuous at: $x = 0, 16, 19$.

Summary of Answers

a)

• $\lim_{x \to 0} f(x) = 2$
• $\lim_{x \to 18^-} f(x) = 6$
• $\lim_{x \to 18^+} f(x) = 4$ (Thus, diverges overall at $x = 18$)

b) The limit exists at $x = 0; 16; 19$.

c) The function is continuous at $x = 0; 16; 19$.

Would you like more details or have any further questions?

5 Relative Questions:

1. How do left-hand and right-hand limits impact overall limit existence?
2. What conditions make a function discontinuous at a certain point?
3. What other types of discontinuities can occur besides jumps?
4. How can graph analysis help determine function behavior at specific points?
5. How does continuity at a point relate to differentiability?

Tip:

Always check left-hand and right-hand limits separately to confirm whether a function is continuous or the overall limit exists.