1. **State the problem:** Determine the truth value of each limit statement about the function $y=f(x)$ based on the graph description. 2. **Analyze each limit:** - a. $\lim_{x \to 2} f(x)$ does not exist. - The graph near $x=2$ has points approaching from the left along the semicircle ending near an open circle at $y=1$, and at $x=2$ the function value is an open circle (not defined). - Since the limit depends on the values from both sides, check if the left and right-hand limits at $2$ are equal. - From the description, there is an arc reaching near $(2,1)$, but the right-hand side (between 2 and 3) likely continues the semicircle symmetrically. - Both sides approach the same value $1$ as $x \to 2$ but the function value at $2$ is undefined. - Therefore, the limit $\lim_{x \to 2} f(x) = 1$ exists, so statement a is FALSE. - b. $\lim_{x \to 2} f(x) = 2.$ - The limit from both sides is $1$ (the height of the semicircle at $x=2$), not $2$. - Statement b is FALSE. - c. $\lim_{x \to 1} f(x)$ does not exist. - From the left of $x=1$, the function is a horizontal line at $y=0$ approaching an open circle at $(1,0)$. - From the right, the semicircle segment starts with an open circle at $(1,0)$. - Both left and right-hand limits approach $0$. - Therefore, the limit at $x=1$ exists and equals $0$. - Statement c is FALSE. - d. $\lim_{x \to x_0} f(x)$ exists at every point $x_0$ in $(-1, 1)$. - On $(-1,0)$, the graph decreases smoothly to $(-1,-1)$. - On $(0,1)$ the graph is a horizontal straight line. - There are no discontinuities or jumps within $(-1,1)$. - Limits exist for all $x_0$ in $(-1,1)$. - Statement d is TRUE. - e. $\lim_{x \to x_0} f(x)$ exists at every point $x_0$ in $(1,3)$. - At $x=2$, the limit exists as shown above. - At $x=3$, there is an open circle indicating no function value; the graph ends there. - The description does not indicate any jump or discontinuity except open circles. - Assuming the semicircle is continuous (except possibly at $x=3$ with an open circle). - For $x_0$ approaching $3$ from the left, limit likely exists (value converging to $0$). - So limits exist for every point in $(1,3)$. - Statement e is TRUE. 3. **Final answers:** - a is FALSE - b is FALSE - c is FALSE - d is TRUE - e is TRUE