## 5.6. Find the limit of the function

$$
\lim _{x \rightarrow \infty} \frac{3 x^{4}-2 x^{3}+6 x^{2}+x-2}{9 x^{4}+5 x^{3}+7 x^{2}+x+1}
$$
Solution.

$$
\begin{gathered}
\lim _{x \rightarrow \infty} \frac{3 x^{4}-2 x^{3}+6 x^{2}+x-2}{9 x^{4}+5 x^{3}+7 x^{2}+x+1}=\lim _{x \rightarrow \infty} \frac{\frac{3 x^{4}-2 x^{3}+6 x^{2}+x-2}{x^{4}}}{\frac{9 x^{4}+5 x^{3}+7 x^{2}+x+1}{x^{4}}}= \\
=\frac{3-2 \lim _{x \rightarrow \infty} \frac{1}{x}+6 \lim _{x \rightarrow \infty} \frac{1}{x^{2}}+\lim _{x \rightarrow \infty} \frac{1}{x^{3}}-2 \lim _{x \rightarrow \infty} \frac{1}{x^{4}}}{9+5 \lim _{x \rightarrow \infty} \frac{1}{x}+7 \lim _{x \rightarrow \infty} \frac{1}{x^{2}}+\lim _{x \rightarrow \infty} \frac{1}{x^{3}}+\lim _{x \rightarrow \infty} \frac{1}{x^{4}}}= \\
=\frac{3-2 \cdot 0+6 \cdot 0+0-2 \cdot 0}{9+5 \cdot 0+7 \cdot 0+0+0}=\frac{1}{3}
\end{gathered}
$$