What possible values can take $a,b\in\mathbb{R}$ such that
$$
\lim\limits_{n\rightarrow\infty}\left( \sqrt[3]{an^{3}+bn^{2}+1}-\log
_{5}\left( 3^{n}+4^{n}+5^{n}\right) -\sqrt[4]{n^{4}+1}\right) =1~?
$$
Denote $\left( a_{n}\right) $ the expresion inside limit. My idea is to
manipulate the expresions one-by-one inside the limit and I get that
\begin{align*}
\lim\limits_{n\rightarrow\infty}\left( a_{n}-1\right) & =\lim
\limits_{n\rightarrow\infty}\left( \sqrt[3]{an^{3}+bn^{2}+1}-1-\log_{5}%
5^{n}\left( \left( \dfrac{3}{5}\right) ^{n}+\left( \dfrac{4}{5}\right)
^{n}+1\right) -\sqrt[4]{n^{4}+1}\right) \\
& =\lim\limits_{n\rightarrow\infty}\left( \sqrt[3]{an^{3}+bn^{2}+1}%
-1-n+\log_{5}\left( \left( \dfrac{3}{5}\right) ^{n}+\left( \dfrac{4}%
{5}\right) ^{n}+1\right) -\sqrt[4]{n^{4}+1}\right) \\
& =\lim\limits_{n\rightarrow\infty}\left( \sqrt[3]{an^{3}+bn^{2}%
+1}-1-n-\sqrt[4]{n^{4}+1}\right) \\
& =\lim\limits_{n\rightarrow\infty}\left( \sqrt[3]{an^{3}+bn^{2}%
+1}-1-2n-\left( \sqrt[4]{n^{4}+1}-n\right) \right) \\
& =\lim\limits_{n\rightarrow\infty}\left( \dfrac{an^{3}+bn^{2}+1-1}
{\sqrt[3]{an^{3}+bn^{2}+1}^{2}+\sqrt[3]{an^{3}+bn^{2}+1}+1}-2n-\lim
\limits_{n\rightarrow\infty}\dfrac{1}{n^{\alpha}+...}\right) \\
& =\lim\limits_{n\rightarrow\infty}\left( \dfrac{an^{3}+bn^{2}}
{\sqrt[3]{an^{3}+bn^{2}+1}^{2}+\sqrt[3]{an^{3}+bn^{2}+1}+1}-2n\right) .
\end{align*}
where I have used $\log_{5}\left( \left( \dfrac{3}{5}\right) ^{n}+\left(
\dfrac{4}{5}\right) ^{n}+1\right) \rightarrow\log_{5}1=0.$
The limit $\lim\limits_{n\rightarrow\infty}\left( a_{n}-1\right) $ should be
equal with $0$, but I do not think that my last relation would imply this
thing. Am I wrong something?
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