If the function f(x)=(tan (tan x)-sin (sin x))/(tan x-sin x) is continuous at x=0, then f(0) is equal to ____________.

Solution

Maths sir · 4 steps

Everything is 0/0 at order x, so expand to the x cubed terms and the answer appears.

1. What continuity needs. For f to be continuous at 0, f(0) must equal lim(x → 0) f(x). Both the numerator and the denominator tend to 0, so you need to look beyond the first-order terms.
2. Expand each piece. Use tan t = t + t³/3 + … and sin t = t - t³/6 + …, with t = tan x or t = sin x. Then tan(tan x) = x + x³/3 + x³/3 = x + 2x³/3 and sin(sin x) = x - x³/6 - x³/6 = x - x³/3, up to x³.

   lim(x→ 0)((x + 2x³/3) - (x - x³/3))/((x + x³/3) - (x - x³/6)) = lim(x→ 0) x³/(x³/2)

3. Cancel x cubed. The numerator is x³ + … and the denominator is x³/2 + …. Dividing gives a limit of 1/(1/2) = 2.
4. So. The limit is 2, so continuity forces f(0) = 2.

The common slip: 1. Keeping only first-order terms, with tan(tan x) ≈ tan x and sin(sin x) ≈ sin x, makes the fraction look like 1. The x³ terms decide the limit.

The idea to keep. When the first-order terms cancel in both the numerator and the denominator, expand to the x³ terms. For nested functions, put the inner series inside the outer one.

Answer from the NTA final answer key. Solution by PrepWiser.