1. Evaluate limits for $g(x) = \frac{-4}{(x-1)^2}$: (a) $\lim_{x \to 1^-} g(x)$: As $x$ approaches 1 from the left, $(x-1)^2$ approaches 0 from the positive side (since square is always positive). So denominator $\to 0^+$, numerator is $-4$, so $g(x) \to -\infty$. (b) $\lim_{x \to 1^+} g(x)$: Similarly, from the right side, denominator $\to 0^+$, so $g(x) \to -\infty$. (c) $\lim_{x \to 1} g(x)$: Both sides go to $-\infty$, so limit is $-\infty$. 2. Evaluate limits for $h(z) = \frac{17}{(4-z)^3}$: (a) $\lim_{z \to 4^-} h(z)$: As $z \to 4^-$, $(4-z) \to 0^+$, so $(4-z)^3 \to 0^+$, so $h(z) \to +\infty$. (b) $\lim_{z \to 4^+} h(z)$: As $z \to 4^+$, $(4-z) \to 0^-$, so $(4-z)^3 \to 0^-$, so $h(z) \to -\infty$. (c) $\lim_{z \to 4} h(z)$: Left and right limits differ ($+\infty$ and $-\infty$), so limit does not exist. 3. Evaluate limits for $g(t) = \frac{4t^2}{(t+3)^7}$: (a) $\lim_{t \to -3^-} g(t)$: $(t+3) \to 0^-$, so denominator $\to 0^-$, numerator $\to 4(-3)^2=36$. Since denominator is very small negative, fraction $\to -\infty$. (b) $\lim_{t \to -3^+} g(t)$: $(t+3) \to 0^+$, denominator $\to 0^+$, numerator $36$, so fraction $\to +\infty$. (c) $\lim_{t \to -3} g(t)$: Left and right limits differ, so limit does not exist. 4. Evaluate limits for $f(x) = \frac{1+x}{x^3+8}$: Note $x^3+8 = (x+2)(x^2 - 2x + 4)$, denominator zero at $x=-2$. (a) $\lim_{x \to -2^-} f(x)$: Denominator $\to 0^-$ (since $x+2 \to 0^-$), numerator $\to -1$, so fraction $\to \frac{-1}{0^-} = +\infty$. (b) $\lim_{x \to -2^+} f(x)$: Denominator $\to 0^+$, numerator $\to -1$, fraction $\to \frac{-1}{0^+} = -\infty$. (c) $\lim_{x \to -2} f(x)$: Left and right limits differ, so limit does not exist. 5. Evaluate limits for $f(x) = \frac{x-1}{(x^2 - 9)^4}$: Note $x^2 - 9 = (x-3)(x+3)$, zero at $x=3$. (a) $\lim_{x \to 3^-} f(x)$: $(x^2-9)^4 \to 0^+$ (even power), numerator $3-1=2$, so fraction $\to +\infty$. (b) $\lim_{x \to 3^+} f(x)$: Similarly, denominator $\to 0^+$, numerator $2$, fraction $\to +\infty$. (c) $\lim_{x \to 3} f(x) = +\infty$. 6. Evaluate limits for $W(t) = \ln(t+8)$: (a) $\lim_{t \to -8^-} W(t)$: $t+8 \to 0^-$, log undefined for negative, so limit does not exist. (b) $\lim_{t \to -8^+} W(t)$: $t+8 \to 0^+$, $\ln(0^+) = -\infty$. (c) $\lim_{t \to -8} W(t)$: Left limit undefined, right limit $-\infty$, so limit does not exist. 7. Evaluate limits for $h(z) = \ln|z|$: (a) $\lim_{z \to 0^-} \ln|z| = \ln(0^+) = -\infty$. (b) $\lim_{z \to 0^+} \ln|z| = \ln(0^+) = -\infty$. (c) $\lim_{z \to 0} \ln|z| = -\infty$. 8. Evaluate limits for $R(y) = \cot(y)$ at $y \to \pi$: Recall $\cot(y) = \frac{\cos y}{\sin y}$, $\sin \pi = 0$. (a) $\lim_{y \to \pi^-} \cot(y)$: $\sin y \to 0^+$, $\cos \pi = -1$, so fraction $\to \frac{-1}{0^+} = -\infty$. (b) $\lim_{y \to \pi^+} \cot(y)$: $\sin y \to 0^-$, fraction $\to \frac{-1}{0^-} = +\infty$. (c) $\lim_{y \to \pi} \cot(y)$ does not exist. 9. Vertical asymptote of $h(x) = \frac{-6}{9-x}$: Denominator zero at $x=9$, so vertical asymptote at $x=9$. 10. Vertical asymptotes of $f(x) = \frac{x+8}{x^2 (5-2x)^3}$: Denominator zero at $x=0$ and $x=\frac{5}{2}$, so vertical asymptotes at $x=0$ and $x=\frac{5}{2}$. 11. Vertical asymptotes of $g(t) = \frac{5t}{t (t+7)(t-12)}$: Simplify numerator and denominator: numerator $5t$, denominator $t(t+7)(t-12)$. Cancel $t$ (except at $t=0$ where function undefined): $$g(t) = \frac{\cancel{5t}}{\cancel{t} (t+7)(t-12)} = \frac{5}{(t+7)(t-12)}$$ Vertical asymptotes at $t=-7$ and $t=12$. At $t=0$ original function undefined but removable discontinuity. 12. Vertical asymptotes of $g(z) = \frac{z^2 + 1}{(z^2 - 1)^5 (z+15)^6}$: Denominator zero at $z^2 -1=0 \Rightarrow z=\pm 1$ and $z=-15$. Vertical asymptotes at $z=-1$, $z=1$, and $z=-15$. Final answers summarized: 1. (a) $-\infty$ (b) $-\infty$ (c) $-\infty$ 2. (a) $+\infty$ (b) $-\infty$ (c) DNE 3. (a) $-\infty$ (b) $+\infty$ (c) DNE 4. (a) $+\infty$ (b) $-\infty$ (c) DNE 5. (a) $+\infty$ (b) $+\infty$ (c) $+\infty$ 6. (a) DNE (b) $-\infty$ (c) DNE 7. (a) $-\infty$ (b) $-\infty$ (c) $-\infty$ 8. (a) $-\infty$ (b) $+\infty$ (c) DNE 9. Vertical asymptote at $x=9$ 10. Vertical asymptotes at $x=0$, $x=\frac{5}{2}$ 11. Vertical asymptotes at $t=-7$, $t=12$ 12. Vertical asymptotes at $z=-1$, $z=1$, $z=-15$