Evaluate lim x→0 (tan x − x)/x^3.

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Multiple Choice

Evaluate lim x→0 (tan x − x)/x^3.

Explanation:
When a limit involves a function at a small input, using its small-angle expansion helps. The Maclaurin series for tan x starts with tan x = x + x^3/3 + higher-order terms. So tan x − x ≈ x^3/3 for small x, with the next terms being of order x^5. Dividing by x^3 gives (tan x − x)/x^3 ≈ 1/3 + terms that vanish as x → 0. Therefore, the limit is 1/3.

When a limit involves a function at a small input, using its small-angle expansion helps. The Maclaurin series for tan x starts with tan x = x + x^3/3 + higher-order terms. So tan x − x ≈ x^3/3 for small x, with the next terms being of order x^5. Dividing by x^3 gives (tan x − x)/x^3 ≈ 1/3 + terms that vanish as x → 0. Therefore, the limit is 1/3.

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