Find $\lim_{x\to0}\frac{\ln(1+\sin^3x \cos^2x)\cot(\ln^3(1+x))\tan^4x}{\sin(\sqrt{x^2+2}-\sqrt{2})\ln(1+x^2)}$ $$\lim_{x\to0}\frac{\ln(1+\sin^3x \cos^2x)\cot(\ln^3(1+x))\tan^4x}{\sin(\sqrt{x^2+2}-\sqrt{2})\ln(1+x^2)}$$
I don't think L'hospital's rule will make the problem easy. (I am afraid to differentiate the numerator). The given limit has a $\frac{0}{0}$ form. I tried using taylor series but the it made the problem more complicated. 

This is exactly the kind of question which "looks very difficult" and at the same time is "extremely easy to answer". It appears that it is specially crafted to generate a complicated looking expression in order to intimidate a casual reader/student. The following evaluation shows that the difficulty is only superficial:
\begin{align}
L &= \lim_{x \to 0}\frac{\log(1 + \sin^{3}x \cos^{2}x)\cot(\log^{3}(1 + x))\tan^{4}x}{\sin(\sqrt{x^{2} + 2} -\sqrt{2})\log(1 + x^{2})}\tag{1}\\
&= \lim_{x \to 0}\dfrac{\log(1 + \sin^{3}x \cos^{2}x)\cot(\log^{3}(1 + x))\cdot\dfrac{\tan^{4}x}{x^{4}}\cdot x^{4}}{\sin(\sqrt{x^{2} + 2} -\sqrt{2})\cdot\dfrac{\log(1 + x^{2})}{x^{2}}\cdot x^{2}}\notag\\
&= \lim_{x \to 0}\dfrac{\log(1 + \sin^{3}x \cos^{2}x)\cot(\log^{3}(1 + x))\cdot 1 \cdot x^{4}}{\sin(\sqrt{x^{2} + 2} -\sqrt{2})\cdot 1 \cdot x^{2}}\notag\\
&= \lim_{x \to 0}\dfrac{\log(1 + \sin^{3}x \cos^{2}x)x^{2}}{\sin(\sqrt{x^{2} + 2} -\sqrt{2})\tan(\log^{3}(1 + x))}\notag\\
&= \lim_{x \to 0}\dfrac{\dfrac{\log(1 + \sin^{3}x \cos^{2}x)}{\sin^{3}x \cos^{2}x}\cdot \sin^{3}x \cos^{2}x\cdot x^{2}}{\sin(\sqrt{x^{2} + 2} -\sqrt{2})\cdot\dfrac{\tan(\log^{3}(1 + x))}{\log^{3}(1 + x)}\cdot \dfrac{\log^{3}(1 + x)}{x^{3}}\cdot x^{3}}\notag\\
&= \lim_{x \to 0}\dfrac{1\cdot \sin^{3}x \cos^{2}x\cdot x^{2}}{\sin(\sqrt{x^{2} + 2} -\sqrt{2})\cdot1\cdot 1\cdot x^{3}}\notag\\
&= \lim_{x \to 0}\dfrac{\sin^{3}x \cdot 1}{\sin(\sqrt{x^{2} + 2} -\sqrt{2})x}\notag\\
&= \lim_{x \to 0}\dfrac{\dfrac{\sin^{3}x}{x^{3}} \cdot x^{3}}{\dfrac{\sin(\sqrt{x^{2} + 2} -\sqrt{2})}{\sqrt{x^{2} + 2} -\sqrt{2}}\cdot(\sqrt{x^{2} + 2} -\sqrt{2})x}\notag\\
&= \lim_{x \to 0}\dfrac{1 \cdot x^{3}}{1\cdot(\sqrt{x^{2} + 2} -\sqrt{2})x}\notag\\
&= \lim_{x \to 0}\frac{x^{2}}{\sqrt{x^{2} + 2} - \sqrt{2}}\tag{2}\\
&= \lim_{x \to 0}\frac{x^{2}(\sqrt{x^{2} + 2} + \sqrt{2})}{(x^{2} + 2) - 2}\notag\\
&= \lim_{x \to 0}(\sqrt{x^{2} + 2} + \sqrt{2})\notag\\
&= \sqrt{2} + \sqrt{2}\notag\\
&= 2\sqrt{2}\notag
\end{align}
The simplification of the complicated expression in equation $(1)$ to a very simple expression in equation $(2)$ is done via the use of standard limits $$\lim_{x \to 0}\cos x = \lim_{x \to 0}\frac{\sin x}{x} = \lim_{x \to 0}\frac{\tan x}{x} = \lim_{x \to 0}\frac{\log(1 + x)}{x} = 1$$ In evaluation of limits it is important to figure out the use of standard limits to simplify a complicated expression.
