The given text appears to be incomplete and lacks a clear mathematical question. However, based on the context, it seems to be related to a function or a curve approaching an axis asymptotically. Let's assume the question is about finding the asymptotes of a given function.
- Identify the function for which we need to find the asymptotes.
- Determine the vertical asymptotes by finding the values of \( x \) that make the denominator zero (if applicable).
- Determine the horizontal asymptotes by analyzing the behavior of the function as \( x \) approaches infinity or negative infinity.
- Determine any oblique asymptotes if the degree of the numerator is one more than the degree of the denominator.
The function under consideration is given by
\[
f(x) = \frac{x^2 - 1}{x - 2}
\]
To find the vertical asymptotes, we set the denominator equal to zero:
\[
x - 2 = 0 \implies x = 2
\]
Thus, there is a vertical asymptote at
\[
\boxed{x = 2}
\]
Next, we determine the horizontal asymptote by evaluating the limit of \( f(x) \) as \( x \) approaches infinity:
\[
\lim_{x \to \infty} f(x) = \infty
\]
This indicates that there is no horizontal asymptote, as the function approaches infinity.
Since the degree of the numerator (2) is one more than the degree of the denominator (1), we can find the oblique asymptote by performing polynomial long division:
\[
f(x) \approx x + 2
\]
Thus, the oblique asymptote is given by
\[
\boxed{y = x + 2}
\]
- Vertical Asymptote: \(\boxed{x = 2}\)
- Horizontal Asymptote: None
- Oblique Asymptote: \(\boxed{y = x + 2}\)
Answered
