<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] Note that by standard limits since
$$\frac{a^x-1}{x}\to \log a$$
we have that
$$\frac{3^x+2^x-2}{4^x+2^x-2}=\frac{\frac{3^x-1}x+\frac{2^x-1}x}{\frac{4^x-1}x+\frac{2^x-1}x}\to\frac{\log 3+\log2}{\log 4+\log 2}=\frac{\log 6}{\log 8}$$
</s>