Evaluate the limit as x approaches 0 of (tan x - sin x) / x^3.

Evaluate the limit as x approaches 0 of (tan x - sin x) / x^3.

• A. 0
• B. 1/2
• C. 1
• D. 3/4

Answer: (B)

When x is very small, use the Taylor expansions for sin and tan to capture the leading terms. sin x is x minus x^3/6 plus higher order terms, and tan x is x plus x^3/3 plus higher order terms. Subtracting, the x terms cancel and the next nonzero term is x^3 times (1/3 + 1/6) = x^3/2. Any higher-order terms vanish when divided by x^3 as x approaches 0. Therefore the limit of (tan x − sin x)/x^3 is 1/2.

When x is very small, use the Taylor expansions for sin and tan to capture the leading terms. sin x is x minus x^3/6 plus higher order terms, and tan x is x plus x^3/3 plus higher order terms. Subtracting, the x terms cancel and the next nonzero term is x^3 times (1/3 + 1/6) = x^3/2. Any higher-order terms vanish when divided by x^3 as x approaches 0. Therefore the limit of (tan x − sin x)/x^3 is 1/2.