Question

Evaluate the limit $ \lim_{n \to \infty} \frac{6^n - 9x - 7^n + 1}{\sqrt{2} - \sqrt{11} + \cos n} $

Hint:

When evaluating limits involving exponential functions, identify the dominant terms and analyze the growth rates of the terms in the numerator and denominator.

Answer 1

We are given the limit expression: \[ \lim_{n \to \infty} \frac{6^n - 9x - 7^n + 1}{\sqrt{2} - \sqrt{11} + \cos n} \] To evaluate this limit, we analyze the behavior of the numerator and denominator as \( n \to \infty \): 
- The terms \( 6^n \) and \( 7^n \) grow exponentially as \( n \) increases, while the terms involving \( x \) and constants become insignificant in comparison. 
Hence, the dominant term in the numerator will be \( 6^n - 7^n \). 
- In the denominator, \( \sqrt{2} - \sqrt{11} + \cos n \) is bounded, since \( \cos n \) oscillates between -1 and 1. 
Therefore, the denominator remains finite. Since the numerator grows exponentially, while the denominator stays bounded, the limit of the expression as \( n \to \infty \) will tend to infinity: \[ \lim_{n \to \infty} \frac{6^n - 9x - 7^n + 1}{\sqrt{2} - \sqrt{11} + \cos n} = \infty \] 
Thus, the value of the limit is \( \infty \).