`"lt"_(n to infty)(x^(n) + y^(n))/(x^(n)- y^(n))`, where `x gt y gt 1` is equal to:

Options

A. 0
B. 1
C. `-1`
D. none

Answer

To solve the limit

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

where \( x > y > 1 \), we can follow these steps:

### Step 1: Factor out \( x^n \)

We start by factoring \( x^n \) from both the numerator and the denominator:

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

### Step 2: Simplify the expression

Now we can simplify the expression:

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

### Step 3: Analyze the limit as \( n \to \infty \)

Since \( y < x \), we know that \( \frac{y}{x} < 1 \). As \( n \) approaches infinity, \( \left(\frac{y}{x}\right)^n \) approaches 0:

\[ \lim_{n \to \infty} \left(\frac{y}{x}\right)^n = 0 \]

### Step 4: Substitute the limit into the simplified expression

Now we substitute this limit back into our expression:

\[ \lim_{n \to \infty} \frac{1 + \left(\frac{y}{x}\right)^n}{1 - \left(\frac{y}{x}\right)^n} = \frac{1 + 0}{1 - 0} = \frac{1}{1} = 1 \]

### Conclusion

Thus, the limit is

\[ \lim_{n \to \infty} \frac{x^n + y^n}{x^n - y^n} = 1 \]