Find $\lim _{x \rightarrow \infty}\left[x \sqrt{\left(x^{2}+1\right)}-x^{2}\right]$

Show that the following sequences are convergent: a) $a_{n}=[\\{1 \cdot 3 \cdot 5 \ldots(2 n-1)\\} /\\{2 \cdot 4 \cdot 6 \ldots(2 n)\\}]$ \(\mathrm{n}=1,2,3, \ldots\) b) \(a_{n}=(1 / 1 !)+(1 / 2 !)+\ldots+(1 / n !)\) \(\mathrm{n}=1,2,3, \ldots\)

Show that: a) $\lim _{n \rightarrow \infty}\left[\left\\{n^{4}+n^{3}-1\right\\} /\left\\{\left(n^{2}+2\right)\left(n^{2}-n-1\right)\right\\}\right]=1$ b) $\lim _{n \rightarrow \infty}\left[1+\left(\mathrm{C} / \mathrm{n}^{2}\right)\right]^{\mathrm{n}}=1$, C a constant.

Prove: a) The sequence of functions \(\left\\{\mathrm{f}_{\mathrm{n}}\right\\}\) defined on \(\mathrm{E}\), converges uniformly on \(E\) if and only if for every \(\varepsilon>0\) there exists an integer \(\mathrm{N}\) such that $\mathrm{m} \geq \mathrm{N}, \mathrm{n} \geq \mathrm{N}, \mathrm{x} \in \mathrm{E}$ implies $1 \mathrm{f}_{\mathrm{n}}(\mathrm{x})-\mathrm{f}_{\mathrm{m}}(\mathrm{x}) \mid \leq \varepsilon$ b) Suppose \(\left\\{f_{n}\right\\}\) is a sequence of functions defined on \(E\), and suppose $1 \mathrm{f}_{\mathrm{n}}(\mathrm{x}) 1 \leq \mathrm{M}_{\mathrm{n}}$ \((\mathrm{x} \in \mathrm{E}, \mathrm{n}=1,2,3, \ldots)\), where $\mathrm{M}_{\mathrm{n}}=\sup _{\mathrm{x} \in \mathrm{E}}\left|\mathrm{f}_{\mathrm{n}}(\mathrm{x})\right|$ and $\lim _{\mathrm{n} \rightarrow \infty} \mathrm{f}_{\mathrm{n}}(\mathrm{x})=0, \mathrm{x} \in \mathrm{E}$, then \({ }^{\infty} \sum_{n=1} f_{n}\) converges uniformly on \(\mathrm{E}\) if \(\sum \mathrm{M}_{\mathrm{n}}\) converges.

Find \(\lim _{n \rightarrow \infty} f_{n}(x) \quad\) where a) \(f_{n}(x)=\left[x /\left(1+n x^{2}\right)\right]\) for \(-1 \leq x \leq 1\) Also find, \(\lim _{n \rightarrow \infty} f_{n}^{\prime}(x)\). b) \(f_{n}(x)=x^{n}\) for \(0 \leq x \leq 1\)

Let \(\left\\{p_{n}\right\\}\) sequence in a metric space \(X\). Prove: a) \(\left\\{p_{n}\right\\}\) converges to \(P \in X\) if and only if every neighborhood of \(p\) contains all but finitely many of the terms of \(\left\\{p_{n}\right\\}\). b) If \(p \in X, q \in X\) and if \(\left\\{p_{n}\right\\}\) converges to \(p\) and to \(q\), then \(q=p\) c) If \(\left\\{p_{n}\right\\}\) converges, then \(\left\\{p_{n}\right\\}\) is bounded. d) If \(E \subset X\) and if \(p\) is a limit point of \(E\), then there is a sequence \(\left\\{p_{n}\right\\}\) in \(E\) such that $$ \mathrm{p}=\lim _{\mathrm{n} \rightarrow \infty} \mathrm{p}_{\mathrm{n}} $$