The given limit expression is: \[ \lim_{n \to \infty} \frac{6^n - 9x - 7^n + 1}{\sqrt{2} - \sqrt{11} + \cos n} \]. To evaluate this limit, we examine the numerator and denominator as \( n \to \infty \):
- In the numerator, \( 6^n \) and \( 7^n \) exhibit exponential growth. The terms \( -9x \) and \( +1 \) are constant relative to \( n \). Thus, the dominant terms are \( 6^n - 7^n \).
- In the denominator, \( \sqrt{2} - \sqrt{11} + \cos n \) is bounded because \( \cos n \) oscillates between -1 and 1, ensuring the denominator remains finite.
As the numerator grows exponentially and the denominator remains bounded, the limit tends to infinity: \[ \lim_{n \to \infty} \frac{6^n - 9x - 7^n + 1}{\sqrt{2} - \sqrt{11} + \cos n} = \infty \].