<s>[INST] Evaluate $\lim_{x\to 0}\frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}$ 
Evaluate
  $$ \displaystyle \lim_{x\to 0}\Bigg( \frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}\Bigg).$$

I tried to use L'Hopital's rule but it got very messy. Moreover I also tried to analyze from graphs, but I was getting the limit $= 0$ by observing it. However, the answer given in my book is $\frac{1}{4}$. Is there any method to do without Taylor series and L' Hopital's rule (like using special limits). We are given that the limit exists. Any help will be appreciated.
Thanks!  
 [/INST] It is difficult to solve the problem without using Taylor series or L'Hospital Rule. Elegant solutions are provided on the assumption that the limit exists (this is also mentioned by OP in his post). I have found another solution which makes minimal use of L'Hospital Rule.
\begin{align}
L &= \lim_{x \to 0}\frac{(\cos x)^{\sin x} - \sqrt{1 - x^{3}}}{x^{6}}\notag\\
&= \lim_{x \to 0}\frac{(\cos x)^{\sin x} - \sqrt{1 - x^{3}}}{x^{6}}\cdot\frac{(\cos x)^{\sin x} + \sqrt{1 - x^{3}}}{(\cos x)^{\sin x} + \sqrt{1 - x^{3}}}\notag\\
&= \frac{1}{2}\lim_{x \to 0}\frac{(\cos x)^{2\sin x} - (1 - x^{3})}{x^{6}}\notag\\
&= \frac{1}{2}\lim_{x \to 0}\frac{\exp(2\sin x\log \cos x) - \exp(\log(1 - x^{3}))}{x^{6}}\notag\\
&= \frac{1}{2}\lim_{x \to 0}\exp(\log(1 - x^{3}))\cdot\frac{\exp(2\sin x\log \cos x - \log(1 - x^{3})) - 1}{x^{6}}\notag\\
&= \frac{1}{2}\lim_{x \to 0}\frac{\exp(2\sin x\log \cos x - \log(1 - x^{3})) - 1}{2\sin x\log \cos x - \log(1 - x^{3})}\cdot\frac{2\sin x\log \cos x - \log(1 - x^{3})}{x^{6}}\notag\\
&= \frac{1}{2}\lim_{x \to 0}1\cdot\frac{2\sin x\log \cos x - \log(1 - x^{3})}{x^{6}}\notag\\
&= \frac{1}{2}\left(\lim_{x \to 0}\frac{2\sin x\log \cos x  + x^{3}}{x^{6}} - \lim_{x \to 0}\frac{x^{3} + \log(1 - x^{3})}{x^{6}}\right)\notag\\
\end{align}
Note that the first limit above is $0$ from this question. Hence we have
\begin{align}
L &= -\frac{1}{2}\lim_{x \to 0}\frac{x^{3} + \log(1 - x^{3})}{x^{6}}\notag\\
&= -\frac{1}{2}\lim_{t \to 0}\frac{t + \log(1 - t)}{t^{2}}\text{ (putting } t = x^{3})\notag\\
&= -\frac{1}{2}\lim_{t \to 0}\dfrac{1 -\dfrac{1}{1 - t}}{2t}\text{ (via LHR)}\notag\\
&= \frac{1}{4}\notag
\end{align}
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