Greater powers

$\displaystyle \lim_{x \to\ \infty} \frac {(2x - 1)^6(3x - 1)^4}{(2x + 1)^{10}} =$

$\displaystyle \lim_{x \to\ \infty} \frac {(2x)^6(3x)^4}{(2x)^{10}} =$

$\displaystyle \lim_{x \to\ \infty} \frac {(2^6)(x^6)(3^4)(x^4)}{(2^{10})(x^{10})} =$

$\displaystyle \lim_{x \to\ \infty} \bigg[ \frac {(2^6)(3^4)}{2^{10}} \times \frac{x^{10}}{x^{10}} \bigg]=$

$\displaystyle \lim_{x \to\ \infty} \frac {3^4}{2^4} =$

$\displaystyle \lim_{x \to\ \infty} \frac {81}{16} =$

$\displaystyle \frac {81}{16}$

$\lim_{x→∞} \frac{(2x-1)^6(3x-1)^4}{(2x+1)^{10}}$

$=\lim_{x→∞} \frac{(2x-1)^6(3x-1)^4}{(2x+1)^6(2x+1)^4}$

$=\lim_{x→∞} \left(\frac{2x-1}{2x+1}\right)^6\times\left(\frac{3x-1}{2x+1}\right)^4$

$=\lim_{x→∞}\left(\frac{x(2 - \frac{6}{x})}{x(2+\frac{1}{x})}\right)^6\times \left(\frac{x(3 - \frac{1}{x})}{x(2+\frac{1}{x})}\right)^4$

$=\left(\frac{2-0}{2+0}\right)^6\times \left(\frac{3-0}{2-0}\right)^4$

$=\boxed{\frac{81}{16}}$

$\begin{aligned} L &= \lim_{x \to \infty} \frac {(2x-1)^6(3x-1)^4}{(2x+1)^{10}} & \small \color{#3D99F6} \text{Divide up and down by }x^{10} \\ &= \lim_{x \to \infty} \frac {\left(2-\frac 1x \right)^6\left(3-\frac 1x \right)^4}{\left(2 +\frac 1x \right)^{10}} \\ & = \frac {2^63^4} {2^{10}} \\ & = \boxed{\dfrac {81}{16}} \end{aligned}$