Sequences

A sequence $(a_n)$ converges to a real number $L$ if for every $\epsilon >0$ there exists an integer $N$ such that $|a_n-L|<\epsilon$ whenever $n\ge N$. We write ${\lim }_{n\to \infty }{a}_n=L$. To evaluate limits, use standard facts: ${\lim }_{n\to \infty }\frac{1}{n^p}=0$ for $p>0$, ${\lim }_{n\to \infty }{\left(1+\frac{x}{n}\right)}^n=e^x$, and ${\lim }_{n\to \infty }{n}^{1/n}=1$. For rational functions, divide numerator and denominator by the highest power of $n$ appearing. The limit superior is the largest subsequential limit, and the limit inferior is the smallest. A convergent sequence is bounded because all but finitely many terms lie within distance $1$ of $L$, and the remaining finite set is bounded.