<s>[INST] Limit as $x$ approaches infinity of a ratio of exponential functions. The problem appears as follows:
Use algebra to solve the following:
$$\lim_{x\to \infty} \frac{2^{3x+2}}{3^{x+3}}$$
The result is infinity which makes intuitive sense but I can’t get it to yield algebraically.
Attempt:
$$\frac{2^{3x+2}}{3^{x+3}} = \frac{2^{3x}2^2}{3^{x}3^3}$$
I’ve decided against continuing writing out my attempt because writing on an iPad, in Latex, is no fun.
The end-result I produce when taking the limit (as $x$ approaches infinity) of the above expression is $\inf / \inf = 1$.
(Do I have to demonstrate that $2^3x$ grows faster than $3^x$ for large $x$?)
 [/INST] Since 
\begin{align*}
\lim_{x\to \infty} \frac{2^{3x+2}}{3^{x+3}} 
&= \lim_{x\to \infty} \frac{2^2 (2^3)^x}{3^3 3^x} \\ 
&= \lim_{x\to \infty} \frac{2^2}{3^3}\left( \frac{8}{3} \right)^x \\
&= \frac{2^2}{3^3} \lim_{x\to \infty}  \left( \frac{8}{3} \right)^x \\
&> \frac{2^2}{3^3} \lim_{x\to \infty}  2^x, \\
 \end{align*}
we know that $\displaystyle{\lim_{x\to \infty}}  2^x=\infty$.
So $\displaystyle{\lim_{x\to \infty}} \frac{2^{3x+2}}{3^{x+3}} =\infty$. 
</s>