The asymptotes of the tangent function, tan x, occur at the values of x where the function is undefined. This happens because the tangent function is defined as the ratio of the sine function to the cosine function:
tan x = sin x / cos x
Since division by zero is undefined, tan x will have vertical asymptotes at the points where cos x equals zero. The cosine function equals zero at the angles:
x = (2n + 1)π/2
where n is any integer (n = 0, ±1, ±2, …). These points are located at:
- π/2
- 3π/2
- -π/2
- -3π/2
Thus, the asymptotes of tan x are vertical lines occurring at these positions on the x-axis. As x approaches these values from the left, tan x tends to negative infinity, and as x approaches from the right, tan x tends to positive infinity. Therefore, we say that tan x has vertical asymptotes at:
x = (2n + 1)π/2, n ∈ Z