Find the limits in a), b), and c) be ow for the function \( f(x)=\frac{4 x}{x-7} \). Use \( -\infty \) and \( \infty \) when appropriate. a) Select the correct choice below and fill in any answer boxes in your choice. A. \[ \lim _{x \rightarrow 7^{-}} f(x)=-\infty \] (S mplify your answer.) B. The limit does not exist and is neither \( -\infty \) nor \( \infty \). b) Select the correct choice balow and fill in any answer boxes in your choice A. \( \lim f(x)= \) \( \square \) \[ x \rightarrow 7^{+} \] (Simplify your answer:) B. The limit does not exist and is neither \( -\infty \) nor \( \infty \). c) Select the correct choice below and fill in any answer boxes in your choice. A. \( \lim _{x \rightarrow 7} f(x)= \) \( \square \) \( x \rightarrow 7 \) (Simplify your answer) B. The limit does not exist and is neither \( -\infty \) nor \( \infty \).
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Let’s dive into these limit evaluations! When approaching the function \( f(x)=\frac{4x}{x-7} \), as \( x \) approaches 7 from the left (\( 7^{-} \)), the denominator approaches zero from the negative side, causing the whole function to dive toward negative infinity. So indeed, the correct choice for a) is: A. \[ \lim_{x \rightarrow 7^{-}} f(x) = -\infty \] Now, when we check what happens as \( x \) approaches 7 from the right (\( 7^{+} \)), the denominator approaches zero from the positive side, and the function shoots up to positive infinity. Therefore, for b), the choice is: A. \[ \lim_{x \rightarrow 7^{+}} f(x) = \infty \] Finally, when we consider the limit as \( x \) approaches 7 from both sides, since \( f(x) \) diverges in separate directions, the limit as \( x \rightarrow 7 \) does not exist: B. \[ \lim_{x \rightarrow 7} f(x) \text{ does not exist}. \] So to summarize: a) A, b) A, c) B!
