The value of `lim_(xtooo) ((2^(x^(n)))e^((1)/(x))-(3^(x^(n)))e^((1)/(x)))/(x^(n))` (where `n in N`) is

Options

A. `e`
B. `e^(2)`
C. `e^(-1)`
D. 1

Explanation

To solve the limit

\[ \lim_{x \to \infty} \frac{(2^{x^n}) e^{\frac{1}{x}} - (3^{x^n}) e^{\frac{1}{x}}}{x^n} \]

we can follow these steps:

### Step 1: Simplify the expression
We can factor out \( e^{\frac{1}{x}} \) from the numerator:

\[ = e^{\frac{1}{x}} \cdot \frac{2^{x^n} - 3^{x^n}}{x^n} \]

### Step 2: Analyze the limit of \( e^{\frac{1}{x}} \)
As \( x \to \infty \), \( e^{\frac{1}{x}} \) approaches \( e^0 = 1 \).

### Step 3: Focus on the remaining limit
Now we need to evaluate:

\[ \lim_{x \to \infty} \frac{2^{x^n} - 3^{x^n}}{x^n} \]

### Step 4: Identify the dominant term
As \( x \to \infty \), \( 3^{x^n} \) grows faster than \( 2^{x^n} \). Thus, we can rewrite the expression:

\[ 2^{x^n} - 3^{x^n} = -3^{x^n} \left(1 - \left(\frac{2}{3}\right)^{x^n}\right) \]

### Step 5: Substitute back into the limit
Now substitute this back into the limit:

\[ \lim_{x \to \infty} \frac{-3^{x^n} \left(1 - \left(\frac{2}{3}\right)^{x^n}\right)}{x^n} \]

### Step 6: Analyze the limit further
As \( x \to \infty \), \( \left(\frac{2}{3}\right)^{x^n} \to 0 \), so:

\[ 1 - \left(\frac{2}{3}\right)^{x^n} \to 1 \]

Thus, the limit simplifies to:

\[ \lim_{x \to \infty} \frac{-3^{x^n}}{x^n} \]

### Step 7: Evaluate the limit
Since \( 3^{x^n} \) grows much faster than \( x^n \), this limit approaches \( -\infty \).

### Step 8: Combine results
Now we combine this with our earlier result for \( e^{\frac{1}{x}} \):

\[ \lim_{x \to \infty} e^{\frac{1}{x}} \cdot \left(-\infty\right) = -\infty \]

### Conclusion
Thus, the final value of the limit is:

\[ \lim_{x \to \infty} \frac{(2^{x^n}) e^{\frac{1}{x}} - (3^{x^n}) e^{\frac{1}{x}}}{x^n} = -\infty \]