Find 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 DNE.)f(x) = tan(πx)
Question
Find 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 DNE.)f(x) = tan(πx)
Solution
The vertical asymptotes of the function f(x) = tan(πx) occur where the function is undefined. The function tan(πx) is undefined where the denominator (cos(πx)) is zero.
The cosine function is zero at odd multiples of π/2. Therefore, we can set the argument of the cosine function (πx) equal to odd multiples of π/2 and solve for x:
πx = (2n+1)π/2
Solving for x gives:
x = (2n+1)/2
So, the vertical asymptotes of the function f(x) = tan(πx) are at x = (2n+1)/2, where n is an integer.
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