\( \lim _ { n \rightarrow \infty } \frac { 1 } { n } + ( - 1 ) ^ { n } \)

The expression \( \frac{1}{n} \) approaches 0 as \( n \) approaches infinity, while the term \( (-1)^n \) oscillates between -1 and 1. This means the limit depends on the behavior of the oscillating term. For even \( n \), it approaches \( \frac{1}{n} + 1 \) (which goes to 1), and for odd \( n \), it approaches \( \frac{1}{n} - 1 \) (which goes to -1). Therefore, the overall limit does not exist due to this oscillation between two bounds.