Evaluate
`Lim_(x to a ) ((x+2)^(5/3)-(a+2)^(5/3))/(x-a)`

Answer

To evaluate the limit

\[ \lim_{x \to a} \frac{(x+2)^{5/3} - (a+2)^{5/3}}{x - a}, \]

we notice that substituting \( x = a \) directly gives us the indeterminate form \( \frac{0}{0} \). Therefore, we can apply L'Hôpital's Rule, which states that if we have an indeterminate form \( \frac{0}{0} \), we can differentiate the numerator and the denominator.

### Step 1: Differentiate the numerator and denominator

1. **Numerator**: Differentiate \( (x+2)^{5/3} \).
- Using the power rule, the derivative is:
\[ \frac{d}{dx}[(x+2)^{5/3}] = \frac{5}{3}(x+2)^{2/3}. \]

2. **Denominator**: Differentiate \( x - a \).
- The derivative is:
\[ \frac{d}{dx}[x - a] = 1. \]

### Step 2: Apply L'Hôpital's Rule

Now we can rewrite the limit using the derivatives we found:

\[ \lim_{x \to a} \frac{(x+2)^{5/3} - (a+2)^{5/3}}{x - a} = \lim_{x \to a} \frac{\frac{5}{3}(x+2)^{2/3}}{1}. \]

### Step 3: Substitute \( x = a \)

Now we substitute \( x = a \) into the limit:

\[ \lim_{x \to a} \frac{5}{3}(x+2)^{2/3} = \frac{5}{3}(a+2)^{2/3}. \]

### Final Answer

Thus, the limit evaluates to:

\[ \frac{5}{3}(a+2)^{2/3}. \]

---