ind the vertical asymptotes (if any) of the graph of the function. (Use \( n \) as an arbitrary Integer If necessary. If an answer does not exist, enter O. On \[ f(x)=5 \tan (\pi x) \] \( =\square \) Need Help? Rendit Watchli

To find the vertical asymptotes of the function \( f(x) = 5 \tan(\pi x) \), we need to determine where the tangent function is undefined. The tangent function has vertical asymptotes at the points where its argument is equal to \( \frac{\pi}{2} + n\pi \), where \( n \) is any integer. 1. **Set the argument of the tangent function equal to the points of discontinuity:** \[ \pi x = \frac{\pi}{2} + n\pi \] 2. **Solve for \( x \):** \[ x = \frac{1}{2} + n \] This means that the vertical asymptotes of the function \( f(x) = 5 \tan(\pi x) \) occur at: \[ x = \frac{1}{2} + n \quad \text{for any integer } n \] Thus, the vertical asymptotes are located at \( x = \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots \) and so on, in both directions. In conclusion, the vertical asymptotes of the function are: \[ \boxed{\frac{1}{2} + n} \] where \( n \) is any integer.