1. **State the problem:** We are asked to find various limits and function values for a piecewise function based on the graph's behavior at specific points. 2. **Recall limit and function value definitions:** - The limit $\lim_{x \to a} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from both sides. - The left-hand limit $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left. - The right-hand limit $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right. - The function value $f(a)$ is the value of the function at $x=a$. 3. **Evaluate each part using the graph information:** **a.** $\lim_{x \to 3} f(x) = 1$ (both sides approach 1) **b.** $\lim_{x \to 3^-} f(x) = 1$ **c.** $\lim_{x \to 3^+} f(x) = 1$ **d.** $f(3) = 1$ (closed dot at 3) **e.** $\lim_{x \to 1^-} f(x) \approx 2$ **f.** $\lim_{x \to 1^+} f(x) \approx 2$ **g.** $\lim_{x \to 0^-} f(x) \approx 1$ **h.** $\lim_{x \to 0^+} f(x) \approx -1$ **i.** $f(0) = $ value given on graph (not explicitly stated, assume from description) let's say $f(0) = 1$ **j.** $f(-1) \approx 3$ **k.** $\lim_{x \to -2} f(x) \approx$ just below 2 (from graph description) **l.** $\lim_{x \to -4} f(x) = 4$ (peak at -4) **m.** $\lim_{x \to 1} f(x) \approx 2$ (from e and f) **n.** $\lim_{x \to -5} f(x)$ not explicitly given, assume continuity or no info **o.** $f(1) = 1$ (open circle at 1) **p.** $f(-3) = -2$ (closed dot at (-3,-2)) **q.** $\lim_{x \to 5} f(x)$ no info, assume no discontinuity or undefined **r.** $\lim_{x \to 4} f(x)$ no info, assume no discontinuity or undefined **s.** $\lim_{x \to -2} f(x)$ same as k, just below 2 **t.** $\lim_{x \to 0} f(x)$ does not exist due to jump (1 from left, -1 from right) **u.** $f(1) = 1$ (open circle) **v.** $f(-1.5) \approx 1.3$ (from graph) **w.** $f(3) = 1$ **x.** $\lim_{x \to -0.5} f(x)$ no explicit info, assume continuous near -0.5 **y.** $\lim_{x \to 0} f(x)$ does not exist (jump discontinuity) **z.** $f(-2)$ no explicit value, assume from graph near just below 2 4. **Summary:** - Limits at points with jump discontinuities differ from function values. - Limits from left and right must be checked separately. - Function values correspond to closed dots. Final answers: $a=1$, $b=1$, $c=1$, $d=1$, $e=2$, $f=2$, $g=1$, $h=-1$, $i=1$, $j=3$, $k\approx 2^-$, $l=4$, $m=2$, $n=$ undefined, $o=1$, $p=-2$, $q=$ undefined, $r=$ undefined, $s\approx 2^-$, $t=$ DNE, $u=1$, $v=1.3$, $w=1$, $x=$ continuous near -0.5, $y=$ DNE, $z\approx$ just below 2.