`lim_(nrarr oo) (a^n_b^n)/(a^n+b^n)`, where `1lt b lt a `, is equal to

Options

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

Explanation

To solve the limit \(\lim_{n \to \infty} \frac{a^n - b^n}{a^n + b^n}\) where \(1 < b < a\), we can follow these steps:

### Step 1: Analyze the limit as \(n\) approaches infinity
As \(n\) approaches infinity, both \(a^n\) and \(b^n\) grow, but since \(a > b\), \(a^n\) will dominate \(b^n\). Thus, we can express the limit as:

\[ \lim_{n \to \infty} \frac{a^n - b^n}{a^n + b^n} \]

### Step 2: Factor out \(a^n\) from both the numerator and the denominator
To simplify the expression, we can factor \(a^n\) out of both the numerator and the denominator:

\[ = \lim_{n \to \infty} \frac{a^n(1 - \frac{b^n}{a^n})}{a^n(1 + \frac{b^n}{a^n})} \]

### Step 3: Simplify the expression
Now, we can cancel \(a^n\) from the numerator and the denominator:

\[ = \lim_{n \to \infty} \frac{1 - \frac{b^n}{a^n}}{1 + \frac{b^n}{a^n}} \]

### Step 4: Evaluate \(\frac{b^n}{a^n}\)
Since \(b < a\), we have \(\frac{b}{a} < 1\). Therefore, as \(n\) approaches infinity, \(\left(\frac{b}{a}\right)^n\) approaches 0:

\[ \lim_{n \to \infty} \frac{b^n}{a^n} = 0 \]

### Step 5: Substitute back into the limit
Substituting this back into our limit gives:

\[ = \frac{1 - 0}{1 + 0} = \frac{1}{1} = 1 \]

### Conclusion
Thus, the limit is:

\[ \lim_{n \to \infty} \frac{a^n - b^n}{a^n + b^n} = 1 \]