Find the limits in \( a), b \) ), and \( c \) ) below for the function \( f(x)=\frac{4 x}{x-8} \). Use \( -\infty \) and \( \infty \) when appropriate. a) Select the correct choice below and fill in any answer boxes in your choice. A. \( \lim _{x \rightarrow 8^{-}} f(x)=\square \) (Simplify your answer ) B. The limit does not exist and is neither \( -\infty \) nor \( \infty \). b) Select the correct choice below and fill in any answer boxes in your choice. A. lim \( f(x)=\square \) \( x \rightarrow 8^{+} \) (Simplify your answer.) B. The limit does not exist and is neither \( -\infty \) nor \( \infty \). Select the correct choice nelow and fill in any answer boxes in your choice. A. lim \( f(x)=\square \) \( x \rightarrow 8 \) (Simplify your answer.) B. The limit does not exist and is neither \( -\infty \) nor \( \infty \).

Let's analyze the limits of the function \( f(x) = \frac{4x}{x - 8} \) as \( x \) approaches 8 from different directions. ### a) \(\lim_{x \to 8^-} f(x)\) **Approach from the left (\( x \to 8^- \))**: - As \( x \) approaches 8 from the left, \( x - 8 \) approaches 0 from the negative side. - Thus, \( \frac{4x}{x - 8} \) becomes \( \frac{32}{\text{negative small number}} \), which tends to \( -\infty \). **Answer**: **A.** \( \lim_{x \to 8^-} f(x) = -\infty \) ### b) \(\lim_{x \to 8^+} f(x)\) **Approach from the right (\( x \to 8^+ \))**: - As \( x \) approaches 8 from the right, \( x - 8 \) approaches 0 from the positive side. - Thus, \( \frac{4x}{x - 8} \) becomes \( \frac{32}{\text{positive small number}} \), which tends to \( \infty \). **Answer**: **A.** \( \lim_{x \to 8^+} f(x) = \infty \) ### c) \(\lim_{x \to 8} f(x)\) **Two-sided approach**: - From part (a), as \( x \) approaches 8 from the left, \( f(x) \) tends to \( -\infty \). - From part (b), as \( x \) approaches 8 from the right, \( f(x) \) tends to \( \infty \). - Since the left-hand limit and right-hand limit are not equal (they approach different infinities), the two-sided limit does not exist. **Answer**: **B.** The limit does not exist and is neither \( -\infty \) nor \( \infty \). --- **Summary of Answers**: - **a.** A. \( \lim_{x \to 8^-} f(x) = -\infty \) - **b.** A. \( \lim_{x \to 8^+} f(x) = \infty \) - **c.** B. The limit does not exist and is neither \( -\infty \) nor \( \infty \).