If `lim_(x to 2) (X^(n) - 2^(n))/(x - 2) = 448` and `n in N` , find n

Answer

To solve the limit problem, we start with the given expression:

\[ \lim_{x \to 2} \frac{x^n - 2^n}{x - 2} = 448 \]

### Step 1: Apply the Limit Formula
We can use the limit formula:

\[ \lim_{x \to a} \frac{x^n - a^n}{x - a} = n \cdot a^{n-1} \]

In our case, we set \( a = 2 \). Thus, we have:

\[ \lim_{x \to 2} \frac{x^n - 2^n}{x - 2} = n \cdot 2^{n-1} \]

### Step 2: Set the Equation
From the limit, we equate it to 448:

\[ n \cdot 2^{n-1} = 448 \]

### Step 3: Simplify the Equation
To find \( n \), we can express 448 in terms of powers of 2:

\[ 448 = 2^6 \cdot 7 \]

### Step 4: Rewrite the Equation
Now we can rewrite our equation:

\[ n \cdot 2^{n-1} = 2^6 \cdot 7 \]

### Step 5: Compare Powers of 2
This implies:

\[ n \cdot 2^{n-1} = 7 \cdot 2^6 \]

From this, we can see that \( n - 1 = 6 \) (the exponent of 2) and \( n = 7 \).

### Step 6: Verify the Solution
Now we substitute \( n = 7 \) back into the equation to verify:

\[ 7 \cdot 2^{7-1} = 7 \cdot 2^6 = 7 \cdot 64 = 448 \]

This confirms that our solution is correct.

### Final Answer
Thus, the value of \( n \) is:

\[ \boxed{7} \]