vertical and horizontal asymptote of arcsin, arccos, and arctan

To find the vertical and horizontal asymptotes of the functions arcsin(x), arccos(x), and arctan(x), we need to consider the properties of these functions.

1. Arcsin(x):
- The domain of arcsin(x) is [-1, 1], which means it is defined for all values of x between -1 and 1.
- The range of arcsin(x) is [-π/2, π/2], which means its output values are bound between -π/2 and π/2.
- As arcsin(x) approaches -1 or 1, the function approaches -π/2 or π/2, respectively. Therefore, these become the horizontal asymptotes.
- Since arcsin(x) is defined for all real numbers, there are no vertical asymptotes.

2. Arccos(x):
- The domain of arccos(x) is also [-1, 1], like arcsin(x), meaning it is defined for all x values between -1 and 1.
- The range of arccos(x) is [0, π], which indicates its output values are bound between 0 and π.
- As arccos(x) approaches -1 or 1, the function approaches π or 0, respectively. Thus, these are the horizontal asymptotes.
- Similar to arcsin(x), there are no vertical asymptotes for arccos(x).

3. Arctan(x):
- The domain of arctan(x) is all real numbers (-∞, +∞), meaning it is defined for any x value.
- The range of arctan(x) is (-π/2, π/2), which implies its output values are bound between -π/2 and π/2.
- As x approaches -∞ or +∞, arctan(x) approaches -π/2 or π/2, respectively. Hence, these are the horizontal asymptotes.
- Similarly, there are no vertical asymptotes for arctan(x).

In summary:
- For arcsin(x), the horizontal asymptotes are y = -π/2 and y = π/2, and there are no vertical asymptotes.
- For arccos(x), the horizontal asymptotes are y = π and y = 0, and there are no vertical asymptotes.
- For arctan(x), the horizontal asymptotes are y = -π/2 and y = π/2, and there are no vertical asymptotes.