The slant asymptote of the function $f(x) = \frac{x^2 + 1}{x}$ is given by the line $y = x$.

To understand how this slant asymptote is formed:

1. Degree Comparison:
   The slant asymptote arises because the degree of the numerator (which is 2, from $x^2$) is exactly one more than the degree of the denominator (which is 1, from $x$).

2. Behavior as $x$ Approaches Infinity or Negative Infinity:

   • As $x$ becomes very large (positively or negatively), the term $\frac{1}{x}$ in $f(x) = x + \frac{1}{x}$ becomes very small and tends to 0.
   • Hence, the function $f(x)$ approaches $y = x$ as $x$ becomes large in magnitude.

Values of $x$ Making the Slant Asymptote

The slant asymptote $y = x$ is not dependent on specific values of $x$; rather, it represents the end behavior of the function as $x$ approaches infinity or negative infinity.

However, you can determine how closely the function aligns with the slant asymptote for specific values of $x$:

• For large positive values of $x$ (e.g., $x = 100$), the function $f(x)$ will be very close to $y = x$ because $\frac{1}{x}$ is very small.
• For large negative values of $x$ (e.g., $x = -100$), the function will again be very close to $y = x$.

In summary, all values of $x$ contribute to the formation of the slant asymptote as $x$ moves towards $\infty$ or $-\infty$. The slant asymptote describes the overall trend rather than being tied to specific $x$ values.

Do you have more questions about asymptotes or how they relate to specific functions? Here are some related questions:

1. How do you find where a function crosses its slant asymptote?
2. What does the remainder term represent in the expression $f(x) = x + \frac{1}{x}$?
3. Can a function have both a slant asymptote and a horizontal asymptote?
4. How would you analyze the behavior of $f(x) = \frac{x^2 + 1}{x}$ near $x = 0$?
5. How do vertical asymptotes differ from slant asymptotes in rational functions?

Tip: When analyzing functions with asymptotes, consider the end behavior of the function by looking at limits as $x$ approaches infinity or negative infinity. This helps in understanding the overall behavior and identifying slant asymptotes.