Evaluate `lim_(x to a) ((x + 4)^(5//4) - (a + 4)^(5//4))/(x - a)`

Explanation

To evaluate the limit

\[ \lim_{x \to a} \frac{(x + 4)^{\frac{5}{4}} - (a + 4)^{\frac{5}{4}}}{x - a}, \]

we can apply the limit property that states:

\[ \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f'(a), \]

where \( f(x) = (x + 4)^{\frac{5}{4}} \).

### Step 1: Identify the function and find its derivative

Let

\[ f(x) = (x + 4)^{\frac{5}{4}}. \]

We need to find \( f'(x) \).

### Step 2: Differentiate \( f(x) \)

Using the power rule of differentiation:

\[ f'(x) = \frac{5}{4}(x + 4)^{\frac{5}{4} - 1} \cdot (1) = \frac{5}{4}(x + 4)^{\frac{1}{4}}. \]

### Step 3: Evaluate the derivative at \( x = a \)

Now, we evaluate \( f'(a) \):

\[ f'(a) = \frac{5}{4}(a + 4)^{\frac{1}{4}}. \]

### Step 4: Substitute back into the limit

Thus, we have:

\[ \lim_{x \to a} \frac{(x + 4)^{\frac{5}{4}} - (a + 4)^{\frac{5}{4}}}{x - a} = f'(a) = \frac{5}{4}(a + 4)^{\frac{1}{4}}. \]

### Final Answer

Therefore, the limit is

\[ \frac{5}{4}(a + 4)^{\frac{1}{4}}. \]

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