\( 1 \leftarrow \) Find the given limit. \( \lim _{x \rightarrow-7} \frac{x^{2}-9}{7-x} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \lim _{x \rightarrow-7} \frac{x^{2}-9}{7-x}=\sqrt{\frac{41}{14}} \) (Simplify your answer.) B. The limit does not exist.

To find the limit \( \lim _{x \rightarrow -7} \frac{x^{2}-9}{7-x} \), we first substitute \(-7\) into the expression: \[ \frac{(-7)^{2} - 9}{7 - (-7)} = \frac{49 - 9}{7 + 7} = \frac{40}{14} = \frac{20}{7} \] Since we have a defined value, the limit definitely exists. So, the correct choice is: The limit exists and is \( \frac{20}{7} \). Now for some additional insights: In the early 17th century, mathematicians began formalizing calculus, particularly around limits. This period, known as the Scientific Revolution, featured contributors like Isaac Newton and Gottfried Wilhelm Leibniz, whose work laid the groundwork for future mathematics. Their discoveries help explain concepts that you use every day, like rates of change and motion! If you encounter a limit that results in an indeterminate form, like \( \frac{0}{0} \), it’s a signal to factor, simplify, or apply L'Hôpital's rule. Always check if the function can be simplified before taking the limit to avoid unnecessary complications. A common mistake is to overlook simplification, leading to a false conclusion that the limit does not exist.