Evaluate the following limit. Use l'Hôpital's Rule when it is convenient and applicable. $$ \lim _{v \rightarrow 7} \frac{v-2-\sqrt{v^{2}-24}}{v-7} $$

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To evaluate the limit
$$
\lim_{v \rightarrow 7} \frac{v - 2 - \sqrt{v^{2} - 24}}{v - 7},
$$
we can use **l'Hôpital's Rule** since substituting $$ v = 7 $$ directly into the expression yields an indeterminate form $$ \frac{0}{0} $$.

**Step 1: Verify the Indeterminate Form**

Substitute $$ v = 7 $$:
$$
\text{Numerator: } 7 - 2 - \sqrt{7^2 - 24} = 5 - \sqrt{49 - 24} = 5 - 5 = 0,
$$
$$
\text{Denominator: } 7 - 7 = 0.
$$
Thus, the limit is indeed of the form $$ \frac{0}{0} $$, and we can apply l'Hôpital's Rule.

**Step 2: Apply l'Hôpital's Rule**

Differentiate the numerator and the denominator with respect to $$ v $$:

- **Numerator:**
  $$
  \frac{d}{dv} \left( v - 2 - \sqrt{v^2 - 24} \right) = 1 - \frac{v}{\sqrt{v^2 - 24}}.
  $$

- **Denominator:**
  $$
  \frac{d}{dv} (v - 7) = 1.
  $$

Thus, the limit becomes:
$$
\lim_{v \rightarrow 7} \left( 1 - \frac{v}{\sqrt{v^2 - 24}} \right).
$$

**Step 3: Evaluate the Limit**

Substitute $$ v = 7 $$ into the derivative expression:
$$
1 - \frac{7}{\sqrt{7^2 - 24}} = 1 - \frac{7}{\sqrt{49 - 24}} = 1 - \frac{7}{5} = 1 - 1.4 = -0.4.
$$
Expressed as a fraction, this is:
$$
-0.4 = -\frac{2}{5}.
$$

**Alternative Verification: Rationalization**

To confirm, we can rationalize the numerator:

$$
\frac{v - 2 - \sqrt{v^2 - 24}}{v - 7} \times \frac{v - 2 + \sqrt{v^2 - 24}}{v - 2 + \sqrt{v^2 - 24}} = \frac{(v - 2)^2 - (v^2 - 24)}{(v - 7)(v - 2 + \sqrt{v^2 - 24})}.
$$

Simplifying the numerator:
$$
(v - 2)^2 - (v^2 - 24) = v^2 - 4v + 4 - v^2 + 24 = -4v + 28 = -4(v - 7).
$$

Thus, the expression simplifies to:
$$
\frac{-4(v - 7)}{(v - 7)(v - 2 + \sqrt{v^2 - 24})} = \frac{-4}{v - 2 + \sqrt{v^2 - 24}}.
$$

Evaluating the limit as $$ v \rightarrow 7 $$:
$$
\frac{-4}{7 - 2 + \sqrt{49 - 24}} = \frac{-4}{5 + 5} = \frac{-4}{10} = -\frac{2}{5}.
$$

**Final Answer:**
$$
-\frac{2}{5}
$$
The limit is $$ -\frac{2}{5} $$.

Evaluate the following limit. Use l’Hôpital’s Rule when it is convenient and applicable.

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