In the interval (-π, π), a vertical asymptote of y = tan x occurs at which x-value?

In the interval (-π, π), a vertical asymptote of y = tan x occurs at which x-value?

• A. -π/2
• B. 0
• C. π/2
• D. π

Answer: (C)

Vertical asymptotes for tan x happen where the function blows up, which is where cos x = 0 because tan x = sin x / cos x becomes undefined there. Cos x equals zero at x = π/2 + kπ. In the interval (-π, π), that gives two points: x = -π/2 and x = π/2. At each of these x-values, the tangent graph shoots off to positive or negative infinity as you approach from either side (the sign flips across the asymptote). So, a vertical asymptote occurs at x = π/2 (and also at x = -π/2) within this interval. If you’re choosing one representative value, π/2 is a valid example, but remember there’s also an asymptote at -π/2.

Vertical asymptotes for tan x happen where the function blows up, which is where cos x = 0 because tan x = sin x / cos x becomes undefined there.

Cos x equals zero at x = π/2 + kπ. In the interval (-π, π), that gives two points: x = -π/2 and x = π/2. At each of these x-values, the tangent graph shoots off to positive or negative infinity as you approach from either side (the sign flips across the asymptote).

So, a vertical asymptote occurs at x = π/2 (and also at x = -π/2) within this interval. If you’re choosing one representative value, π/2 is a valid example, but remember there’s also an asymptote at -π/2.