1. **State the problem:** We are given the rational function $$f(x) = \frac{5}{x+2} - 2$$ and a table of $x$ values: 0.5, 3, 48, 998, 4998. We need to find the corresponding $f(x)$ values and identify the asymptote. 2. **Formula and rules:** The function is a rational function with a vertical asymptote where the denominator is zero, i.e., at $x+2=0 \Rightarrow x=-2$. The horizontal asymptote is the value that $f(x)$ approaches as $x \to \pm \infty$. 3. **Calculate $f(x)$ for each $x$:** - For $x=0.5$: $$f(0.5) = \frac{5}{0.5+2} - 2 = \frac{5}{2.5} - 2 = 2 - 2 = 0$$ - For $x=3$: $$f(3) = \frac{5}{3+2} - 2 = \frac{5}{5} - 2 = 1 - 2 = -1$$ - For $x=48$: $$f(48) = \frac{5}{48+2} - 2 = \frac{5}{50} - 2 = 0.1 - 2 = -1.9$$ - For $x=998$: $$f(998) = \frac{5}{998+2} - 2 = \frac{5}{1000} - 2 = 0.005 - 2 = -1.995$$ - For $x=4998$: $$f(4998) = \frac{5}{4998+2} - 2 = \frac{5}{5000} - 2 = 0.001 - 2 = -1.999$$ 4. **Identify the asymptote:** - Vertical asymptote at $x = -2$ because the denominator is zero there. - Horizontal asymptote is $y = -2$ because as $x \to \infty$, $\frac{5}{x+2} \to 0$ and $f(x) \to -2$. 5. **Summary:** - Table of values: | $x$ | 0.5 | 3 | 48 | 998 | 4998 | |---|---|---|---|---|---| | $f(x)$ | 0 | -1 | -1.9 | -1.995 | -1.999 | - Asymptotes: - Vertical: $x = -2$ - Horizontal: $y = -2$ The function approaches $y = -2$ as $x$ becomes very large.