What are the values of x in making the slant asymptote of the given function 

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.

Discover how slant asymptotes form in rational functions through an in-depth analysis of the function f(x) = (x^2 + 1) / x. Learn about the degree comparison between the numerator and denominator, and explore the function's behavior as x approaches infinity or negative infinity. Suitable for students in Grades 10-12.

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