<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] $$\frac{5^{x+1}+7^{x+2}}{5^x+7^{x+1}}=\frac{5\left(\frac57\right)^x+7^2}{\left(\frac57\right)^x+7}$$
Now $\displaystyle\lim_{n\to\infty}r^n=0$ if $|r|<1$
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