<s>[INST] Evaluate $\lim_{x\to\infty}\dfrac{5^{x+1}+7^{x+2}}{5^x + 7^{x+1}}$ Evaluate $$\lim_{x\to\infty}\dfrac{5^{x+1}+7^{x+2}}{5^x + 7^{x+1}}$$
I'm getting a different result but not the exact one.
I got $$\dfrac{5\cdot\dfrac{5^n}{7^n} +49}{\dfrac{5^n}{7^n} + 7}.$$
I know the result is $7$ but I cannot figure out the steps.
 [/INST] The powers of $5$ are negligible compared to the powers of $7$ when $x$ is big, so you have in effect $7^{x+2}/7^{x+1}$, so the limit is $7$.
One can write
$$
\lim_{x\to\infty} \frac{5^{x+1}+7^{x+2}}{5^x + 7^{x+1}} = \lim_{x\to\infty}\frac{\left(\frac 5 7\right)^{x+1} + 7}{\frac 1 7\left(\frac 5 7 \right)^x + 1} = \frac{0+7}{0+1}
$$
(since the powers of $5/7$ go to $0$ as $x\to\infty$).
</s>