If the function f(x) = tan(tanx) − sin(sinx) / tanx − sinx is continuous at x = 0, then f(0) is equal to:
❓ Question: If the function f ( x ) = tan ( tan x ) − sin ( sin x ) tan x − sin x is continuous at x = 0 x=0 , then find the value of f ( 0 ) f(0) . 🖼️ Question Image ✍️ Short Solution At x = 0 x=0 : tan ( tan 0 ) = tan 0 = 0 , sin ( sin 0 ) = sin 0 = 0. \tan(\tan 0) = \tan 0 = 0,\quad \sin(\sin 0)=\sin 0=0. So numerator = 0 − 0 = 0 0-0=0 Denominator = tan 0 − sin 0 = 0 − 0 = 0. \tan 0 - \sin 0 = 0-0=0. ⇒ Indeterminate form 0 / 0 0/0 . We must evaluate the limit as x → 0 x\to0 Apply series expansion around x = 0 x=0 : tan x = x + x 3 3 + O ( x 5 ) . \tan x = x + \tfrac{x^3}{3} + O(x^5). sin x = x − x 3 6 + O ( x 5 ) . \sin x = x - \tfrac{x^3}{6} + O(x^5). Hence, tan x − sin x = ( x + x 3 3 ) − ( x − x 3 6 ) + O ( x 5 ) = x 3 2 + O ( x 5 ) . \tan x - \sin x = \Big(x+\tfrac{x^3}{3}\Big) - \Big(x-\tfrac{x^3}{6}\Big) + O(x^5) = \tfrac{x^3}{2} + O(x^5). Expand numerator: tan ( tan x ) ≈ tan ( x + x 3 3 ) . \ta...