Evaluate the following limits :
`Lim_(x to a) ((x+2)^(3//2) -(a+2)^(3//2))/(x-a)`

Answer

To evaluate the limit

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

we can follow these steps:

### Step 1: Substitute the limit
First, we substitute \( x = a \) into the expression:

\[ \frac{(a+2)^{3/2} - (a+2)^{3/2}}{a - a} = \frac{0}{0}. \]

This gives us an indeterminate form \( \frac{0}{0} \).

**Hint:** When you encounter a \( \frac{0}{0} \) form, consider using L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that we can take the derivative of the numerator and the derivative of the denominator separately.

The numerator is \( (x+2)^{3/2} - (a+2)^{3/2} \) and the denominator is \( x - a \).

### Step 3: Differentiate the numerator
The derivative of the numerator \( (x+2)^{3/2} \) is:

\[ \frac{d}{dx}[(x+2)^{3/2}] = \frac{3}{2}(x+2)^{1/2} \cdot \frac{d}{dx}(x+2) = \frac{3}{2}(x+2)^{1/2} \cdot 1 = \frac{3}{2}(x+2)^{1/2}. \]

The derivative of the constant \( (a+2)^{3/2} \) is \( 0 \).

### Step 4: Differentiate the denominator
The derivative of the denominator \( x - a \) is:

\[ \frac{d}{dx}(x - a) = 1. \]

### Step 5: Rewrite the limit
Now we can rewrite the limit using the derivatives:

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

### Step 6: Substitute \( x = a \) again
Now we substitute \( x = a \):

\[ \frac{3}{2}(a+2)^{1/2}. \]

### Final Result
Thus, the limit evaluates to:

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

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