<s>[INST] Find $\lim _{x \to \infty}\frac{5^x+7^{x+1}}{7^x+5^{x+1}}$ 
I just started studying and in our first maths-exam we had to find the following limit:

$\lim \limits_{x \to \infty}\frac{5^x+7^{x+1}}{7^x+5^{x+1}}$

The other tasks where rather easy, so I feel like I'm missing something obvious here, but I don't even really know how to start with this one.
A walkthrough would be great, but hints in the right direction would also help.
Thank you!
 [/INST] I would divide both numerator and denominator by $7^{x+1}$:
\begin{align*}
\frac{5^x+7^{x+1}}{7^x+5^{x+1}} &= \frac{\frac{5^x}{7^{x+1}} + \frac{7^{x+1}}{7^{x+1}}}{\frac{7^x}{7^{x+1}} + \frac{5^{x+1}}{7^{x+1}}}\\
&= \frac{\frac{1}{7}\left(\frac{5}{7}\right)^x + 1}{\frac{1}{7} + \left(\frac{5}{7}\right)^{x+1}}
\end{align*}
Since $\frac{5}{7}<1$, you know what $\left(\frac{5}{7}\right)^x$ tend to as $x\to\infty$.
</s>