Trigonometrical Limits

$\begin{aligned} L & = \lim_{x \to 0} \frac {1-(f(x))^3}{55\sin^2 x} & \small \color{#3D99F6} \text{A 0/0 case, L'Hôpital's rule applies.} \\ & = \lim_{x \to 0} \frac {-3(f(x))^2f'(x)}{110\sin x\cos x} & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }x. \\ & = \lim_{x \to 0} - \frac {3(f(x))^2}{55\sin (2x)} \sum_{k=1}^5 \frac {- 2k \sin (2kx) f(x)}{\cos (2kx)} \\ & = \lim_{x \to 0} \frac {6(f(x))^3}{55\sin (2x)} \sum_{k=1}^5 \frac {k \color{#3D99F6}\sin (2kx)}{\cos (2kx)} & \small \color{#3D99F6} \text{Note that }\sin (2kx) = \sin (2x + 2(k-1)x) \\ & = \lim_{x \to 0} \frac {6(f(x))^3}{55\sin (2x)} \sum_{k=1}^5 k \left( \frac {\color{#3D99F6}\sin (2x)\cos(2(k-1)x) + \sin(2(k-1)x)\cos(2x)}{\cos (2kx)} \right) \\ & = \lim_{x \to 0} \frac {6(f(x))^3}{55\sin (2x)} \sum_{k=1}^5 k \left( \frac {\sin (2x)\cos(2(k-1)x) + (\sin(2kx)\cos(2x)-\sin(2x)\cos(2kx))\cos(2x)}{\cos (2kx)} \right) \\ & = \lim_{x \to 0} \frac {6(f(x))^3}{55} \sum_{k=1}^5 k \left( \frac {\cos(2(k-1)x)}{\cos (2kx)} + \frac {\tan(2kx)\cos^2(2x)}{\sin(2x)}-\cos(2x) \right) \\ & = \lim_{x \to 0} \frac {6(f(x))^3}{55} \sum_{k=1}^5 k \left( \frac {\cos(2(k-1)x)}{\cos (2kx)} + \frac {\tan(2kx)\left(1-\sin^2(2x)\right)}{\sin(2x)}-\cos(2x) \right) \\ & = \lim_{x \to 0} \frac {6(f(x))^3}{55} \sum_{k=1}^5 k \left( \frac {\cos(2(k-1)x)}{\cos (2kx)} + \frac {\tan(2kx)}{\sin(2x)}- \tan(2kx)\sin(2x)-\cos(2x) \right) \\ & = \lim_{x \to 0} \frac {6(f(x))^3}{55} \sum_{k=1}^5 k \left( \frac {\cos(2(k-1)x)}{\cos (2kx)} - \tan(2kx)\sin(2x)-\cos(2x) \right) + \color{#3D99F6} \lim_{x \to 0} \frac {6(f(x))^3}{55} \sum_{k=1}^5 \frac {k\tan(2kx)}{\sin(2x)} & \small \color{#3D99F6} \text{A 0/0 case, L'Hôpital's rule applies.} \\ & = \frac 6{55} \sum_{k=1}^5 k \left(1 - 0 -1 \right) + \color{#3D99F6} \lim_{x \to 0} \frac {6(f(x))^3}{55} \sum_{k=1}^5 k \left(\frac {2k\sec^2(2kx)}{2\cos(2x)} \right) & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }x. \\ & = 0 + \frac {6}{55} \sum_{k=1}^5 k^2 = \frac 6{55} \times \frac {5(6)(11)}6 = \boxed{6} \end{aligned}$