Find the value of n, if `lim_(xto2) (x^(n)-2^(n))/(x-2)=80`, `n in N`.

Options

A. `3`
B. `5`
C. `4`
D. `6`

Explanation

To solve the limit problem, we need to find the value of \( n \) such that:

\[ \lim_{x \to 2} \frac{x^n - 2^n}{x - 2} = 80 \]

### Step 1: Recognize the limit form
We can use the standard limit result:

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

In our case, \( a = 2 \).

### Step 2: Apply the standard limit result
Substituting \( a = 2 \) into the standard limit formula, we have:

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

### Step 3: Set the limit equal to 80
According to the problem, this limit equals 80:

\[ n \cdot 2^{n-1} = 80 \]

### Step 4: Simplify the equation
We can express 80 in terms of powers of 2:

\[ 80 = 5 \cdot 16 = 5 \cdot 2^4 \]

Thus, we rewrite the equation:

\[ n \cdot 2^{n-1} = 5 \cdot 2^4 \]

### Step 5: Equate the powers of 2
From the equation \( n \cdot 2^{n-1} = 5 \cdot 2^4 \), we can compare coefficients:

1. The coefficient \( n \) must equal 5.
2. The powers of 2 give us \( n - 1 = 4 \).

### Step 6: Solve for \( n \)
From \( n - 1 = 4 \):

\[ n = 5 \]

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

\[ \boxed{5} \]