for either case using formula on wiki, I'm stuck at these limits (putting $\partial Z_0=1$): $\lim_{t \to \infty} \frac{\ln(-sint dt)}{t}$ and for the other one $\lim_{t \to \infty} \frac{\ln(3 dt)}{t}$
Sorry? What is, in the second case, say, $3dt$ when $t\to\infty$? And why put $\partial Z_0=1$, as you say, when the limits are considered for $\delta Z_0\to0$?
What does the WP page you link to, actually say? One considers a dynamical system, that is, in continuous time $Z'(t)=f(Z(t))$ or in discrete time $Z_{n+1}=g(Z_n)$, and one is interested in
the speed at which trajectories starting from two different points diverge one from the another when the time is large.
To make this precise in the discrete setting, one considers two starting points $x$ and $y$ and the systems $Z^x_{n+1}=g(Z^x_n)$, $Z^x_0=x$, and $Z^y_{n+1}=g(Z^y_n)$, $Z^y_0=y$. If $Z^x_n-Z^y_n$ behaves roughly as $\mathrm e^{n\lambda}$, then one calls $\lambda$ the Lyapunov exponent.
Of course, more care is needed to define this rigorously: one actually fixes some $x$ and one considers the limit $y\to x$, then, if $|Z^y_n-Z^x_n|/|y-x|\to k_n(x)$ when $y\to x$ and if $k_n(x)=\mathrm e^{n\lambda(x)+o(n)}$ when $n\to\infty$, then $\lambda(x)$ is the Lyapunov exponent at $x$. Thus, $$\lambda(x)=\lim_{n\to\infty}\frac1n\lim_{y\to x}\log\left(\frac{|Z^y_n-Z^x_n|}{|y-x|}\right).$$
This may seem abstruse but the idea is simple, so let us see what happens when $g:z\mapsto3z$ (your second case). Then $Z^x_n=3^nx$ for every $x$ and every $n$ hence $$\frac{|Z^y_n-Z^x_n|}{|y-x|}=3^n,$$ for every $y\ne x$, in particular $k_n(x)=3^n$ for every $n$ and indeed $\lambda(x)$ exists for every $x$ and $\lambda(x)=3$.
Likewise, if $g:z\mapsto\cos(z)$ and $x=\cos(x)$, that is, $x\approx0.739$ is the Dottie number, then, for every $y$, $Z^y_n=\cos^{\circ n}(y)$ hence $k_n(x)=|(\cos^{\circ n})'(x)|$. For every $z$, $$(\cos^{\circ n})'(z)=(-\sin z)(-\sin(\cos z))\cdots(-\sin(\cos^{\circ (n-1)} z)),$$ in particular $k_n(x)=(\sin x)^n$ hence $\lambda(x)=\log\sin x\approx-0.395$.
