Questions: lim as x approaches 9 of (sqrt(x^2 - 5x - 36))/(8 - 3x)

Solution Steps

To solve the limit \(\lim _{x \rightarrow 9} \frac{\sqrt{x^{2}-5 x-36}}{8-3 x}\), we can follow these steps:

1. Simplify the Expression: Factorize the quadratic expression inside the square root if possible.
2. Substitute the Limit Value: Substitute \(x = 9\) directly into the simplified expression to see if it results in an indeterminate form.
3. Apply Limit Theorems: If direct substitution results in an indeterminate form, apply limit theorems such as L'Hôpital's Rule or algebraic manipulation to resolve the indeterminate form.

We start with the limit: \[ \lim _{x \rightarrow 9} \frac{\sqrt{x^{2}-5 x-36}}{8-3 x} \] First, we simplify the expression inside the square root: \[ x^{2} - 5x - 36 = (x - 9)(x + 4) \] Thus, we can rewrite the limit as: \[ \lim _{x \rightarrow 9} \frac{\sqrt{(x - 9)(x + 4)}}{8 - 3x} \]

Next, we substitute \(x = 9\) into the expression: \[ \frac{\sqrt{(9 - 9)(9 + 4)}}{8 - 3 \cdot 9} = \frac{\sqrt{0}}{8 - 27} = \frac{0}{-19} \] This results in an indeterminate form of \(\frac{0}{-19}\).

Since the limit evaluates to \(0\), we conclude that: \[ \lim _{x \rightarrow 9} \frac{\sqrt{x^{2}-5 x-36}}{8-3 x} = 0 \]

Final Answer