(1) Use the definition of convergence of sequences to verify the following limits: \text { (a) } \lim _{n \rightarrow \infty} \frac{(-1)^{n} n}{n^{2}+1}=0 \text { (b) } \lim _{n \rightarrow

\infty} n\left(\sqrt{1+\frac{1}{n}}-1\right)=\frac{1}{2} \text { . }