Evaluate `lim_(x to 4) (x^(5) - 1024)/(x - 5)`

Explanation

To evaluate the limit \( \lim_{x \to 4} \frac{x^5 - 1024}{x - 5} \), we can follow these steps:

### Step 1: Recognize the expression
We start with the limit:
\[ \lim_{x \to 4} \frac{x^5 - 1024}{x - 5} \]
Notice that \( 1024 = 4^5 \), so we can rewrite the expression as:
\[ \lim_{x \to 4} \frac{x^5 - 4^5}{x - 5} \]

### Step 2: Apply the limit
As \( x \) approaches 4, the numerator \( x^5 - 4^5 \) approaches \( 0 \) and the denominator \( x - 5 \) approaches \( -1 \). However, we can simplify the numerator using the difference of powers formula.

### Step 3: Use the difference of powers formula
The difference of powers can be factored as:
\[ x^5 - 4^5 = (x - 4)(x^4 + x^3 \cdot 4 + x^2 \cdot 4^2 + x \cdot 4^3 + 4^4) \]
Thus, we rewrite our limit:
\[ \lim_{x \to 4} \frac{(x - 4)(x^4 + x^3 \cdot 4 + x^2 \cdot 4^2 + x \cdot 4^3 + 4^4)}{x - 5} \]

### Step 4: Substitute \( x = 4 \)
Now, we can substitute \( x = 4 \) into the limit:
\[ \lim_{x \to 4} \frac{(x - 4)(x^4 + x^3 \cdot 4 + x^2 \cdot 4^2 + x \cdot 4^3 + 4^4)}{x - 5} \]
As \( x \) approaches 4, the term \( (x - 4) \) goes to 0, but we need to evaluate the remaining expression:
\[ x^4 + x^3 \cdot 4 + x^2 \cdot 4^2 + x \cdot 4^3 + 4^4 \]
Substituting \( x = 4 \):
\[ 4^4 + 4^3 \cdot 4 + 4^2 \cdot 16 + 4 \cdot 64 + 256 \]
Calculating each term:
- \( 4^4 = 256 \)
- \( 4^3 \cdot 4 = 64 \cdot 4 = 256 \)
- \( 4^2 \cdot 16 = 16 \cdot 16 = 256 \)
- \( 4 \cdot 64 = 256 \)
- \( 4^4 = 256 \)

Adding these together:
\[ 256 + 256 + 256 + 256 + 256 = 1280 \]

### Step 5: Final limit evaluation
Now we can evaluate the limit:
\[ \lim_{x \to 4} \frac{(x - 4) \cdot 1280}{x - 5} \]
Since \( x - 4 \) approaches 0 and \( x - 5 \) approaches -1, we can conclude:
\[ \lim_{x \to 4} \frac{0 \cdot 1280}{-1} = 0 \]

### Final Answer
Thus, the limit is:
\[ \boxed{1280} \]