2. Determine the intersection of the graph of the function $f(x)=\frac{2 x^{3}+x+1}{x^{3}+1}$ with its horizontal asymptote.
2. The horizontal asymptote of the given function is $y=2$. To find the intersection, we need to equate $f(x)$ with the asymptote. We get the equation $\frac{2 x^{3}+x+1}{x^{3}+1}=2$. We solve the equation by eliminating the fraction and considering the condition $x \neq-1$. We get the equation $2 x^{3}+x+1=2 x^{3}+2$, which we solve. The solution is $x=1$. The ordinate of the intersection point is the value of the asymptote, so $P(1,2)$.

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-13.jpg?height=51&width=1639&top_left_y=1274&top_left_x=206)

The equation written is $f(x)=2$ or $\frac{2 x^{3}+x+1}{x^{3}+1}=2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-13.jpg?height=52&width=1639&top_left_y=1373&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-13.jpg?height=51&width=1639&top_left_y=1422&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-13.jpg?height=57&width=1642&top_left_y=1471&top_left_x=207)