If the rational function y = r(x) has the vertical asymptote x = 9, then as x
→9+, either y → ____or _____
consider the graph of y = 1/(x-9)
what do you think?
If a rational function has a vertical asymptote at x = 9, it means that as x approaches 9 from the right side, the values of y will either approach positive infinity or negative infinity. Thus, as x approaches 9+ (from the right side), either y approaches positive infinity (y → +∞) or y approaches negative infinity (y → -∞).
If the rational function y = r(x) has a vertical asymptote at x = 9, it means that as x approaches 9 from the right side (x → 9+), y will approach either positive infinity (∞) or negative infinity (-∞).
To understand this, you can follow these steps:
1. Identify the rational function: In this case, the function is y = r(x).
2. Determine the vertical asymptote: The given information states that the vertical asymptote is at x = 9.
3. Understand the behavior as x approaches 9 from the right side: To find out what happens to y as x → 9+ (x approaches 9 from the right side), you need to analyze the behavior of the function.
- If the function approaches positive infinity as x → 9+, it means that the function's values increase without bound as x gets closer to 9 from the right side. This can occur if the function has a vertical asymptote at x = 9 and has a positive slope before reaching x = 9. Symbolically, y will approach ∞.
- If the function approaches negative infinity as x → 9+, it means that the function's values decrease without bound as x gets closer to 9 from the right side. This can occur if the function has a vertical asymptote at x = 9 and has a negative slope before reaching x = 9. Symbolically, y will approach -∞.
In summary, as x → 9+ (x approaches 9 from the right side), either y → ∞ (positive infinity) or y → -∞ (negative infinity).