1. Evaluate limits for $g(x) = \frac{-4}{(x-1)^2}$: (a) As $x \to 1^-$, $(x-1)^2$ approaches 0 from the positive side since squaring any real number is positive. So, $g(x) = \frac{-4}{(x-1)^2} \to \frac{-4}{0^+} = -\infty$. (b) As $x \to 1^+$, similarly $(x-1)^2 \to 0^+$, so $g(x) \to -\infty$. (c) Since both one-sided limits are $-\infty$, $\lim_{x \to 1} g(x) = -\infty$. 2. Evaluate limits for $h(z) = \frac{17}{(4-z)^3}$: (a) As $z \to 4^-$, $(4-z) \to 0^+$, so $(4-z)^3 \to 0^+$. Thus, $h(z) = \frac{17}{0^+} = +\infty$. (b) As $z \to 4^+$, $(4-z) \to 0^-$, so $(4-z)^3 \to 0^-$. Thus, $h(z) = \frac{17}{0^-} = -\infty$. (c) Since one-sided limits differ, $\lim_{z \to 4} h(z)$ does not exist. 3. Evaluate limits for $g(t) = \frac{4t^2}{(t+3)^7}$: (a) As $t \to -3^-$, $(t+3) \to 0^-$, so $(t+3)^7 \to 0^-$ (odd power preserves sign). Numerator $4t^2$ is positive. So, $g(t) = \frac{\text{positive}}{0^-} = -\infty$. (b) As $t \to -3^+$, $(t+3) \to 0^+$, so $(t+3)^7 \to 0^+$. So, $g(t) = \frac{\text{positive}}{0^+} = +\infty$. (c) Since one-sided limits differ, $\lim_{t \to -3} g(t)$ does not exist. 4. Evaluate limits for $f(x) = \frac{1+x}{x^3 + 8}$: Note $x^3 + 8 = (x+2)(x^2 - 2x + 4)$. (a) As $x \to -2^-$, numerator $1 + x \to -1$ (negative), denominator $x^3 + 8 \to 0^-$ (since $x+2 \to 0^-$ and quadratic positive). So, $f(x) = \frac{-1}{0^-} = +\infty$. (b) As $x \to -2^+$, numerator $1 + x \to -1$ (negative), denominator $x^3 + 8 \to 0^+$. So, $f(x) = \frac{-1}{0^+} = -\infty$. (c) Since one-sided limits differ, $\lim_{x \to -2} f(x)$ does not exist. 5. Evaluate limits for $f(x) = \frac{x-1}{(x^2 - 9)^4}$: Note $x^2 - 9 = (x-3)(x+3)$. (a) As $x \to 3^-$, numerator $x-1 \to 2$ (positive), denominator $(x^2 - 9)^4 \to 0^+$ (even power). So, $f(x) = \frac{2}{0^+} = +\infty$. (b) As $x \to 3^+$, numerator $x-1 \to 2$ (positive), denominator $\to 0^+$. So, $f(x) = +\infty$. (c) Since both sides $+\infty$, $\lim_{x \to 3} f(x) = +\infty$. 6. Evaluate limits for $W(t) = \ln(t+8)$: (a) As $t \to -8^-$, $t+8 \to 0^-$, but $\ln$ undefined for negative arguments. So, $\lim_{t \to -8^-} W(t)$ does not exist. (b) As $t \to -8^+$, $t+8 \to 0^+$, so $\ln(t+8) \to -\infty$. (c) Since left limit does not exist, $\lim_{t \to -8} W(t)$ does not exist. 7. Evaluate limits for $h(z) = \ln|z|$: (a) As $z \to 0^-$, $|z| \to 0^+$, so $\ln|z| \to -\infty$. (b) As $z \to 0^+$, same as above, $\ln|z| \to -\infty$. (c) Both sides $-\infty$, so $\lim_{z \to 0} h(z) = -\infty$. 8. Evaluate limits for $R(y) = \cot(y)$ at $y \to \pi$: Recall $\cot(y) = \frac{\cos y}{\sin y}$. (a) As $y \to \pi^-$, $\sin y \to 0^+$, $\cos y \to -1$. So, $\cot(y) = \frac{-1}{0^+} = -\infty$. (b) As $y \to \pi^+$, $\sin y \to 0^-$, $\cos y \to -1$. So, $\cot(y) = \frac{-1}{0^-} = +\infty$. (c) Since one-sided limits differ, $\lim_{y \to \pi} R(y)$ does not exist. 9. Find vertical asymptotes of $h(x) = \frac{-6}{9 - x}$: Denominator zero at $9 - x = 0 \Rightarrow x = 9$. Vertical asymptote at $x=9$. 10. Find vertical asymptotes of $f(x) = \frac{x + 8}{x^2 (5 - 2x)^3}$: Denominator zero when $x^2 = 0 \Rightarrow x=0$ or $5 - 2x = 0 \Rightarrow x=\frac{5}{2}$. Vertical asymptotes at $x=0$ and $x=\frac{5}{2}$. 11. Find vertical asymptotes of $g(t) = \frac{5t}{t (t + 7) (t - 12)}$: Denominator zero at $t=0$, $t=-7$, $t=12$. Note numerator $5t$ zero at $t=0$ as well, so cancel factor $t$: $$g(t) = \frac{\cancel{5t}}{\cancel{t} (t+7)(t-12)} = \frac{5}{(t+7)(t-12)}$$ So no vertical asymptote at $t=0$ (removable discontinuity). Vertical asymptotes at $t=-7$ and $t=12$. 12. Find vertical asymptotes of $g(z) = \frac{z^2 + 1}{(z^2 - 1)^5 (z + 15)^6}$: Denominator zero when $z^2 - 1 = 0 \Rightarrow z=\pm 1$ or $z + 15 = 0 \Rightarrow z = -15$. Vertical asymptotes at $z = -15, -1, 1$. Final answers: 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 $z=-15$, $z=-1$, $z=1$