Consider the function f(x) = (sin(2x) - 2*tan(x)) / x³ for x not equal to 0, and f(0) = k. For what value of k is f continuous at x = 0?

1
-1
2
-2

Correct answer: -2

Solution

Numerator: sin(2x) - 2*tan(x) = (2x - 4x³/3 +...) - (2x + 2x³/3 +...) = -2x³ + higher order terms. Dividing by x³: limit as x->0 = -2. Hence k = -2.

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