Evaluate $\lim_{x\to\infty}\dfrac{5^{x+1}+7^{x+2}}{5^x + 7^{x+1}}$

Question

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.

Answer 1

$$\frac{5^{x+1}+7^{x+2}}{5^x+7^{x+1}}=\frac{5\left(\frac57\right)^x+7^2}{\left(\frac57\right)^x+7}$$

Now $\displaystyle\lim_{n\to\infty}r^n=0$ if $|r|<1$

Answer 2

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 $x\to\infty$).