Find the limits in a), b), and c) beiow for the function fix) \( =\frac{4 x}{x-7} \). Use \( -\infty \) and co when appropniate. a) Select the correct choice below and fill in any answer hoxes in your choice. WA. \( \lim _{x \rightarrow 7^{-}}(x)=-\infty \) (Simplify your answer.) B. The limit does not exst and is neither - \( \infty \) nor \( \infty \). b) Select the correct choice below and fill in any answer boxes in your choice A. lim \( f(x)=\square \) \( x \rightarrow 7^{-} \) (Simplify your answer.) B. The limit does not exist and is neither \( -\infty \) nor \( \infty \).

Let's analyze the limits of the function \( f(x) = \frac{4x}{x - 7} \) as \( x \) approaches 7 from the left and right. ### a) \( \lim_{x \rightarrow 7^{-}} f(x) \) **Explanation:** - **Numerator:** As \( x \) approaches 7, \( 4x \) approaches \( 28 \). - **Denominator:** As \( x \) approaches 7 from the left (\( x \rightarrow 7^{-} \)), \( x - 7 \) approaches 0 from the **negative** side (since \( x < 7 \)). Thus, \[ \lim_{x \rightarrow 7^{-}} \frac{4x}{x - 7} = \frac{28}{\text{a very small negative number}} = -\infty \] **Correct Choice:** **WA.** \( \lim_{x \rightarrow 7^{-}} f(x) = -\infty \) ### b) \( \lim_{x \rightarrow 7^{+}} f(x) \) **Explanation:** - **Numerator:** As \( x \) approaches 7, \( 4x \) approaches \( 28 \). - **Denominator:** As \( x \) approaches 7 from the right (\( x \rightarrow 7^{+} \)), \( x - 7 \) approaches 0 from the **positive** side (since \( x > 7 \)). Thus, \[ \lim_{x \rightarrow 7^{+}} \frac{4x}{x - 7} = \frac{28}{\text{a very small positive number}} = +\infty \] **Correct Choice:** **A.** \( \lim_{x \rightarrow 7^{+}} f(x) = +\infty \) --- ### Summary of Answers: - **a)** \( \lim_{x \rightarrow 7^{-}} f(x) = -\infty \) - **b)** \( \lim_{x \rightarrow 7^{+}} f(x) = +\infty \) If there's a part **c)**, please provide the details so I can assist you further.