Limit Identities

Relevant wiki: L'Hopital's Rule - Basic

$\begin{aligned} L & = \lim_{x\to0} \left [ \frac{2\cos^2 \left( \frac 12 x^4 \right)}{\frac1{18}x} - \frac{18\sin^2\left( \frac 12 x^4 \right) + 36}{x}\right ] \\ & = \lim_{x \to 0} \frac {36\cos^2 \left( \frac 12 x^4 \right) - 18 \sin^2 \left( \frac 12 x^4 \right) - 36}x \\ & = \lim_{x \to 0} \frac {18(2\cos^2 \left( \frac 12 x^4 \right) -1) + 9(1 - 2 \sin^2 \left( \frac 12 x^4 \right)) - 27}x \\ & = \lim_{x \to 0} \frac {18\cos (x^4) + 9\cos (x^4) - 27}x \\ & = \lim_{x \to 0} \frac {27\cos (x^4) - 27}x \quad \quad \small \color{#3D99F6}{\text{This is a 0/0 case, L'Hôpital's rule applies.}} \\ & = \lim_{x \to 0} \frac {-27\sin (x^4)(4x^3)}1 \quad \quad \small \color{#3D99F6}{\text{Differentiate up and down w.r.t. }x.} \\ & = \boxed{0} \end{aligned}$