Problem solving limit infinity/infinity. I cannot solve this limit:
$$\lim_{x\rightarrow\infty}
 \frac {(3x^2-4)  \left( \sqrt [3]{2x^2+1}+1
 \right)^2}{ (2x-1) \left( 4-\sqrt {8x^3-2}
 \right)x^{3/2}}$$
I make this:
$${\frac { \left( 3\,{x}^{2}-4 \right)  \left( \sqrt [3]{2\,{x}^{2}+1}+1
 \right) ^{2}}{ \left( 2\,x-1 \right)  \left( 4-\sqrt {8\,{x}^{3}-2}
 \right) {x}^{3/2}}} =3\,{\frac {\sqrt {x} \left( 2\,{x}^{2}+1 \right) ^{2/3}}{ \left( 2\,x-
1 \right)  \left( 4-\sqrt {8\,{x}^{3}-2} \right) }}-4\,{\frac {
 \left( 2\,{x}^{2}+1 \right) ^{2/3}}{ \left( 2\,x-1 \right)  \left( 4-
\sqrt {8\,{x}^{3}-2} \right) {x}^{3/2}}}+6\,{\frac {\sqrt {x}\sqrt [3]
{2\,{x}^{2}+1}}{ \left( 2\,x-1 \right)  \left( 4-\sqrt {8\,{x}^{3}-2}
 \right) }}-8\,{\frac {\sqrt [3]{2\,{x}^{2}+1}}{ \left( 2\,x-1
 \right)  \left( 4-\sqrt {8\,{x}^{3}-2} \right) {x}^{3/2}}}+3\,{\frac 
{\sqrt {x}}{ \left( 2\,x-1 \right)  \left( 4-\sqrt {8\,{x}^{3}-2}
 \right) }}-4\,{\frac {1}{ \left( 2\,x-1 \right)  \left( 4-\sqrt {8\,{
x}^{3}-2} \right) {x}^{3/2}}}$$
then
$$\lim_{x\rightarrow\infty}3\,{\frac {\sqrt {x} \left( 2\,{x}^{2}+1 \right) ^{2/3}}{ \left( 2\,x-
1 \right)  \left( 4-\sqrt {8\,{x}^{3}-2} \right) }}
$$
$$\lim_{x\rightarrow\infty}-4\,{\frac { \left( 2\,{x}^{2}+1 \right) ^{2/3}}{ \left( 2\,x-1
 \right)  \left( 4-\sqrt {8\,{x}^{3}-2} \right) {x}^{3/2}}}
$$
$$\lim_{x\rightarrow\infty}6\,{\frac {\sqrt {x}\sqrt [3]{2\,{x}^{2}+1}}{ \left( 2\,x-1 \right) 
 \left( 4-\sqrt {8\,{x}^{3}-2} \right) }}
$$
$$\lim_{x\rightarrow\infty}-8\,{\frac {\sqrt [3]{2\,{x}^{2}+1}}{ \left( 2\,x-1 \right)  \left( 4-
\sqrt {8\,{x}^{3}-2} \right) {x}^{3/2}}}
$$
$$\lim_{x\rightarrow\infty}3\,{\frac {\sqrt {x}}{ \left( 2\,x-1 \right)  \left( 4-\sqrt {8\,{x}^{
3}-2} \right) }}
$$
$$\lim_{x\rightarrow\infty}-4\,{\frac {1}{ \left( 2\,x-1 \right)  \left( 4-\sqrt {8\,{x}^{3}-2}
 \right) {x}^{3/2}}}
$$
but I cannot solve this
NOTE: I cannot use L'hopital for finding this limit.

The total power of $x$ in the numerator is $2 + 4/3 = 10/3.$ The total power of $x$ in the denominator is $1 + 3/2 + 3/2 = 4.$ The denominator wins: The limit must be $0.$
