Questions: Evaluate the limit as n approaches infinity of (cos(n)) / (2n + 1).

Solution Steps

To evaluate the limit \(\lim _{n \rightarrow \infty} \frac{\cos n}{2 n+1}\), we need to consider the behavior of the numerator and the denominator as \(n\) approaches infinity. The cosine function oscillates between -1 and 1, while the denominator \(2n + 1\) grows without bound. As the denominator grows much faster than the bounded numerator, the fraction approaches 0.

We are given the limit expression: \[ \lim _{n \rightarrow \infty} \frac{\cos n}{2n + 1} \]

The numerator \(\cos n\) oscillates between \(-1\) and \(1\) for all \(n\).

The denominator \(2n + 1\) increases without bound as \(n\) approaches infinity.

Since the numerator is bounded and the denominator grows without bound, the fraction \(\frac{\cos n}{2n + 1}\) approaches \(0\) as \(n\) approaches infinity.

Final Answer