Final answer:
The function f(x) = (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ) has no horizontal or vertical asymptotes.
Step-by-step explanation:
The given function is f(x) = (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ). To find the asymptotes of this function, we need to determine the values of x for which the function approaches infinity or negative infinity. As x approaches infinity or negative infinity, the terms eˣ and e⁻ˣ become dominant, and the function behaves like eˣ / eˣ = 1. Therefore, there are no horizontal asymptotes.
However, to find the vertical asymptotes, we need to solve the equation eˣ + e⁻ˣ = 0. Let's solve it:
- Subtracting e⁻ˣ from both sides of the equation gives eˣ = -e⁻ˣ.
- Raising both sides to the power of eˣ gives e²ˣ = -1.
- This equation has no real solutions because the exponential function is always positive.
Therefore, the function f(x) = (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ) has no vertical asymptotes.