Jika lim(x → 0) [(1 − cos 2x)/(cos ax − cos bx)] = 1/2, maka a² − b² = ⋯ (A) 8 (B) 2 (C) −2 (D) −4 (E) −8

Jawaban: E Konsep: * 1 – cos 2x = 2 sin² x * cos A - cos B = -2 sin ((A + B)/2) sin ((A - B)/2) * lim (x → 0) (sin ax)/(bx) = a/b * (a + b)(a - b) = a² − b² Pembahasan: lim (x → 0) [(1 − cos 2x)/(cos ax − cos bx)] = 1/2 ⇔ lim (x → 0) [(2 sin² x)/(-2 sin ((ax + bx)/2) sin ((ax - bx)/2))] = 1/2 ⇔ lim (x → 0) [(2 sin² x)/(-2 sin ((a + b)x/2) sin ((a - b)x/2))] = 1/2 ⇔ lim (x → 0) -[(sin² x)/(sin ((a + b)x/2) sin ((a - b)x/2))] = 1/2 ⇔ lim (x → 0) -[(sin x sin x)/(sin ((a + b)x/2) sin ((a - b)x/2))] = 1/2 ⇔ lim (x → 0) -[((x · x)/(((a + b)x/2) · ((a - b)x/2))) · (sin x)/x · (sin x)/x · ((a + b)x/2)/(sin ((a + b)x/2)) · ((a - b)x/2)/(sin ((a - b)x/2)))] = 1/2 ⇔ -[(1/(((a + b)/2) · ((a - b)/2))) · 1 · 1 · 1 · 1] = 1/2 ⇔ -1/(((a + b)/2) · ((a - b)/2))) = 1/2 ⇔ -2 = ((a + b)/2) · ((a - b)/2))) ⇔ -2 = ((a + b)(a - b))/4 ⇔ -8 = (a + b)(a - b) ⇔ (a + b)(a - b) = -8 ⇔ a² − b² = -8 Jadi, nilai a² − b² adalah -8 Oleh karena itu, jawaban yang benar adalah E.