Evaluate `lim_(x to 16) (x^(3//2) - 64)/(x - 16)`

Explanation

To evaluate the limit

\[ \lim_{x \to 16} \frac{x^{3/2} - 64}{x - 16}, \]

we can start by rewriting \(64\) in terms of \(16\):

1. **Rewrite 64**:
\[ 64 = 16^{3/2}. \]
Therefore, we can rewrite the limit as:
\[ \lim_{x \to 16} \frac{x^{3/2} - 16^{3/2}}{x - 16}. \]

2. **Identify the form**:
As \(x\) approaches \(16\), both the numerator and denominator approach \(0\). This is an indeterminate form of type \(\frac{0}{0}\).

3. **Apply the limit formula**:
We can use the limit formula:
\[ \lim_{x \to a} \frac{x^n - a^n}{x - a} = n \cdot a^{n-1}. \]
Here, \(n = \frac{3}{2}\) and \(a = 16\).

4. **Calculate \(n \cdot a^{n-1}\)**:
- First, calculate \(a^{n-1}\):
\[ n - 1 = \frac{3}{2} - 1 = \frac{1}{2}. \]
Thus,
\[ a^{n-1} = 16^{1/2} = 4. \]
- Now, substitute back into the formula:
\[ n \cdot a^{n-1} = \frac{3}{2} \cdot 4 = 6. \]

5. **Final result**:
Therefore, we have:
\[ \lim_{x \to 16} \frac{x^{3/2} - 64}{x - 16} = 6. \]

### Summary of Steps:
1. Rewrite \(64\) as \(16^{3/2}\).
2. Recognize the limit is of the form \(\frac{0}{0}\).
3. Apply the limit formula for derivatives.
4. Calculate \(n \cdot a^{n-1}\).
5. Conclude with the limit value.