Evaluate lim x→0 (tan x − x)/x^3.

Evaluate lim x→0 (tan x − x)/x^3.

• A. 1
• B. 0
• C. -1/3
• D. 1/3

Answer: (D)

When a limit involves a trigonometric function near zero, using its small-angle (Maclaurin) expansion is the most straightforward path. For tan x near 0, the expansion is tan x = x + x^3/3 + higher-order terms. Subtracting the linear term x leaves tan x − x = x^3/3 + higher-order terms. Dividing by x^3 gives (tan x − x)/x^3 = 1/3 + terms that vanish as x → 0. So the limit is 1/3. If you only kept the linear term you’d get 0, which shows why the cubic term matters here. The result is 1/3.

When a limit involves a trigonometric function near zero, using its small-angle (Maclaurin) expansion is the most straightforward path. For tan x near 0, the expansion is tan x = x + x^3/3 + higher-order terms. Subtracting the linear term x leaves tan x − x = x^3/3 + higher-order terms. Dividing by x^3 gives (tan x − x)/x^3 = 1/3 + terms that vanish as x → 0. So the limit is 1/3. If you only kept the linear term you’d get 0, which shows why the cubic term matters here. The result is 1/3.