<s>[INST] 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.
 [/INST] $\lim_\limits{x\rightarrow\infty}
 {\frac { \left( 3\,{x}^{2}-4 \right)  \left( \sqrt [3]{2\,{x}^{2}+1}+1
 \right) ^{2}}{ \left( 2\,x-1 \right)  \left( 4-\sqrt {8\,{x}^{3}-2}
 \right) {x}^{3/2}}}$
$\lim_\limits{x\rightarrow\infty}
 {\frac { \left( 3\,{x}^{2}-4 \right)  \left( (2x^2+1)^{1/3}+1
 \right) ^{2}}{ \left( 2\,x-1 \right)  \left( 4-(8x^3-2)^{1/2}
 \right) {x}^{3/2}}}$
find the highest power of x in the numerator and the denominator.
$\lim_\limits{x\to\infty}
 \frac{3*2^{2/3}x^{10/3} + ax^n + bx^m... }{2*8^{3/2}x^4+cx^{p}+dx^q..} $
with $m <n < 10/3$ and $p<q < 4$
Divide top and bottom by $x^4$
$\lim_\limits{x\to\infty}
 \frac{3*2^{2/3}x^{-2/3} + ax^{n-4} + bx^{m-4}... }{2*8^{3/2}+cx^{p-4}+dx^{q-4}...} $
as x goes to infinity all the terms with x to a negative power go to 0, leaving a zero in the numerator and a constant in the denominator.
$\lim_\limits{x\to\infty}
 \frac{3*2^{2/3}x^{-2/3} + ax^{n-4} + bx^{m-4}... }{2*8^{3/2}+cx^{p-4}+dx^{q-4}...}=0 $
</s>