<s>[INST] Compute the limit $\lim_{x\to 0} \frac{3^x+2^x-2}{4^x+2^x-2}$ 
Compute the limit $\lim_{x\to 0} \frac{3^x+2^x-2}{4^x+2^x-2}$

Here is what I have done so far:
\begin{align}
\lim_{x\to 0} \frac{3^x+2^x-2}{4^x+2^x-2}
&=
\lim_{x\to 0} \left(\frac{3^x-1}{4^x+2^x-2}+\frac{2^x-1}{4^x+2^x-2}\right)\\
&=\lim_{x\to 0} \left(\frac{3^x-1}{x}\frac{x}{4^x+2^x-2}+\frac{2^x-1}{x}\frac{x}{4^x+2^x-2}\right)\\
&=\ln 3\lim_{x\to 0} \frac{x}{4^x+2^x-2}+\ln 2\lim_{x\to 0} \frac{x}{4^x+2^x-2}
\end{align}
 [/INST] \begin{align}\lim_{x\to 0} \frac{3^x+2^x-2}{4^x+2^x-2}&=\lim_{x\to 0} \left(\frac{3^x-1}{4^x+2^x-2}+\frac{2^x-1}{4^x+2^x-2}\right)
\\&=\lim_{x\to 0} \left(\frac{3^x-1}{x}\frac{x}{4^x+2^x-2}+\frac{2^x-1}{x}\frac{x}{4^x+2^x-2}\right)
\\&=\ln 3\lim_{x\to 0} \frac{x}{4^x+2^x-2}+\ln 2\lim_{x\to 0} \frac{x}{4^x+2^x-2} \\&= (\ln 3 + \ln 2) \lim_{x\to 0} \frac{x}{4^x+2^x-2} 
\\&= (\ln 3 + \ln 2) \lim_{x\to 0} \frac{1}{\frac{4^x-1}{x}+\frac{2^x-1}{x}} \\
&=\frac{\ln 3 + \ln 2}{\ln 4 + \ln 2} \\
&= \frac{\ln 6}{\ln 8}
\end{align}
</s>