Determine the two-sided limit lim x-> pi/2 tan x.

Determine the two-sided limit lim x-> pi/2 tan x.

• A. DNE (left: infinity, right: -infinity)
• B. +∞
• C. -∞
• D. 0

Answer: (A)

The behavior near π/2 is governed by tan x = sin x / cos x. As x approaches π/2 from the left, cos x → 0+ while sin x → 1, so tan x → 1/(0+) = +∞. As x approaches π/2 from the right, cos x → 0− while sin x → 1, so tan x → 1/(0−) = −∞. Since the left-hand limit goes to +∞ and the right-hand limit goes to −∞, the two-sided limit does not exist. This matches the description of a vertical asymptote at π/2 with opposite directions on either side.

The behavior near π/2 is governed by tan x = sin x / cos x. As x approaches π/2 from the left, cos x → 0+ while sin x → 1, so tan x → 1/(0+) = +∞. As x approaches π/2 from the right, cos x → 0− while sin x → 1, so tan x → 1/(0−) = −∞. Since the left-hand limit goes to +∞ and the right-hand limit goes to −∞, the two-sided limit does not exist. This matches the description of a vertical asymptote at π/2 with opposite directions on either side.