If `0lt xlty`, then `lim_(xrarroo) (y^n+x^n)^(1//n)` is equal to

Options

A. e
B. x
C. y
D. none of these

Answer

To solve the limit \( \lim_{x \to \infty} (y^n + x^n)^{1/n} \) where \( 0 < y < x \), we can follow these steps:

### Step-by-Step Solution

1. **Identify the Limit Expression**:
We start with the expression:
\[ L = \lim_{x \to \infty} (y^n + x^n)^{1/n} \]

2. **Factor Out \( x^n \)**:
Since \( x \) is approaching infinity and \( y \) is a constant, we can factor \( x^n \) out of the expression inside the limit:
\[ L = \lim_{x \to \infty} (x^n(1 + \frac{y^n}{x^n}))^{1/n} \]

3. **Simplify the Expression**:
We can simplify the expression:
\[ L = \lim_{x \to \infty} (x^n)^{1/n} \cdot (1 + \frac{y^n}{x^n})^{1/n} \]
This simplifies to:
\[ L = \lim_{x \to \infty} x \cdot (1 + \frac{y^n}{x^n})^{1/n} \]

4. **Evaluate the Limit of the Second Part**:
As \( x \to \infty \), \( \frac{y^n}{x^n} \) approaches 0. Therefore:
\[ (1 + \frac{y^n}{x^n})^{1/n} \to (1 + 0)^{1/n} = 1 \]

5. **Combine the Results**:
Now we can combine the results:
\[ L = \lim_{x \to \infty} x \cdot 1 = \lim_{x \to \infty} x = \infty \]

### Final Result
Thus, the limit is:
\[ L = \infty \]