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 \( \infty \).
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To find the limits of \( f(x) = \frac{4x}{x-8} \) at \( x = 8 \), we need to analyze the behavior as \( x \) approaches 8 from the left and the right. a) A. To find \( \lim _{x \rightarrow 8^{-}} f(x) \), as \( x \) approaches 8 from the left, the denominator \( x - 8 \) approaches 0 from the negative side, which causes \( f(x) \) to tend toward \( -\infty \). Therefore, the limit is: \( \lim _{x \rightarrow 8^{-}} f(x) = -\infty \). b) A. For \( \lim _{x \rightarrow 8^{+}} f(x) \), as \( x \) approaches 8 from the right, the denominator \( x - 8 \) approaches 0 from the positive side, which causes \( f(x) \) to tend toward \( +\infty \). Hence, the limit is: \( \lim _{x \rightarrow 8^{+}} f(x) = +\infty \). c) A. Since the left-hand limit approaches \( -\infty \) and the right-hand limit approaches \( +\infty \), we conclude that: \( \lim _{x \rightarrow 8} f(x) \) does not exist. So the final answers are: a) \( \lim _{x \rightarrow 8^{-}} f(x) = -\infty \) b) \( \lim _{x \rightarrow 8^{+}} f(x) = +\infty \) c) \( \lim _{x \rightarrow 8} f(x) \) does not exist.
