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Proof: $\frac{n^{2}+5}{n+1} \rightarrow \infty$ as $n \rightarrow \infty$
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Proof. Let $a_{n}:=\frac{n^{2}+5}{n+1}$ and let $A>0$ be given. Observe that

\begin{aligned} a_{n}:=\frac{n^{2}+5}{n+1} & \geq \frac{n^{2}}{n+1} \\ & \geq \frac{n^{2}}{n+n} \\ &=\frac{n^{2}}{2 n} \\ &=\frac{n}{2} \end{aligned}

and $\frac{n}{2}>A$ provided that $n>2 A$.
So, let $N$ be any natural number larger that $2 A$. Then if $n>N$, we have $a_{n}>\frac{n}{2}>\frac{N}{2}>A$. Therefore, $a_{n}$ tends to infinity.
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