117. Find $\lim _{x \rightarrow \infty} \frac{2 x^{3}+x}{x^{3}-1}$.
Solution. Here, the numerator and denominator do not have a limit, as both increase indefinitely. In this case, it is said that there is an indeterminate form of $\infty / \infty$. We will divide the numerator and denominator term by term by $x^{3}$ (the highest power of $x$ in this fraction):

$$
\lim _{x \rightarrow \infty} \frac{2 x^{3}+x}{x^{3}-1}=\lim _{x \rightarrow \infty} \frac{2+\frac{1}{x^{2}}}{1-\frac{1}{x^{3}}}=\frac{\lim _{x \rightarrow \infty}\left(2+\frac{1}{x^{2}}\right)}{\lim _{x \rightarrow \infty}\left(1-\frac{1}{x^{3}}\right)}=2
$$

since $1 / x^{2}$ and $1 / x^{3}$ tend to zero as $x \rightarrow \infty$.

Remark. The method of dividing the numerator and denominator of a fraction by the highest power of the variable $x$, used to resolve indeterminate forms of $\infty / \infty$, cannot be used to find the limits of functions that do not lead to the indeterminate form of the specified type.