Answer :
To determine the horizontal asymptote of the function \( f(x) = \frac{-2}{x-2} - 2 \), we need to analyze the behavior of the function as \( x \) approaches infinity (or negative infinity). The horizontal asymptote will be the value that the function \( f(x) \) approaches as \( x \) becomes very large (positively or negatively).
Let's examine the given function \( f(x) = \frac{-2}{x-2} - 2 \).
1. Consider the term \(\frac{-2}{x-2}\):
- As \( x \) becomes very large (i.e., \( x \to \infty \)), the term \( x - 2 \) also becomes very large.
- Since \(\frac{-2}{x-2}\) involves division by a large number, \(\frac{-2}{x-2}\) approaches 0.
2. Analyze the behavior as \( x \to \infty \):
[tex]\[ \lim_{{x \to \infty}} f(x) = \lim_{{x \to \infty}} \left( \frac{-2}{x-2} - 2 \right) \][/tex]
- As established, \(\frac{-2}{x-2} \to 0\) as \( x \to \infty \).
- Hence, we get:
[tex]\[ \lim_{{x \to \infty}} f(x) = 0 - 2 = -2 \][/tex]
Therefore, as \( x \) approaches infinity, \( f(x) \) approaches \(-2\).
3. Conclusion:
The horizontal asymptote of the function \( f(x) = \frac{-2}{x-2} - 2 \) is \( y = -2 \). In other words, \( f(x) \) approaches the value \(-2\) as \( x \) becomes very large in either direction.
Thus, the correct answer is:
[tex]\[ \boxed{f(x) = -2} \][/tex]
Let's examine the given function \( f(x) = \frac{-2}{x-2} - 2 \).
1. Consider the term \(\frac{-2}{x-2}\):
- As \( x \) becomes very large (i.e., \( x \to \infty \)), the term \( x - 2 \) also becomes very large.
- Since \(\frac{-2}{x-2}\) involves division by a large number, \(\frac{-2}{x-2}\) approaches 0.
2. Analyze the behavior as \( x \to \infty \):
[tex]\[ \lim_{{x \to \infty}} f(x) = \lim_{{x \to \infty}} \left( \frac{-2}{x-2} - 2 \right) \][/tex]
- As established, \(\frac{-2}{x-2} \to 0\) as \( x \to \infty \).
- Hence, we get:
[tex]\[ \lim_{{x \to \infty}} f(x) = 0 - 2 = -2 \][/tex]
Therefore, as \( x \) approaches infinity, \( f(x) \) approaches \(-2\).
3. Conclusion:
The horizontal asymptote of the function \( f(x) = \frac{-2}{x-2} - 2 \) is \( y = -2 \). In other words, \( f(x) \) approaches the value \(-2\) as \( x \) becomes very large in either direction.
Thus, the correct answer is:
[tex]\[ \boxed{f(x) = -2} \][/tex]
