$$\lim_{x \rightarrow \infty}\frac{5^{x+1}+7^{x+1}}{5^x +7^x}$$
I tried using L'Hospital rule, which yielded :
$$\lim_{x \rightarrow \infty}\frac{5^{x+1} \ln 5+7^{x+1} \ln 7}{5^x \ln 5 }$$
But I'm at the dead end… If I divide numerator and denominator by $5^x$ , I get a term $\frac{7^x}{5^x}$ … which is unsolvable for the limit $x \rightarrow \infty $
However , the answer provided by book is $-7$ . I doubt there is mistake in the question.
Best Answer
The standard method is to factor out and cancel the highest power, which is here $7^x$:
We get:
$\lim_{x\to\infty} \frac{5^{x+1}+7^{x+1}}{5^x+7^x}=\lim_{x\to\infty} \frac{7^{x+1}\left(\left(\frac57\right)^{x+1}+1\right)}{7^x\left(\left(\frac57\right)^x+1\right)}=\lim_{x\to\infty} \frac{7\left(\left(\frac57\right)^{x+1}+1\right)}{\left(\frac57\right)^x+1}$
$=7\cdot\lim_{x\to\infty} \frac{\left(\frac57\right)^{x+1}+1}{\left(\frac57\right)^x+1}$
It is $\lim_{x\to\infty} \left(\frac{5}{7}\right)^x=0$, since $\frac57<1$
We get:
$=7\cdot\frac{0+1}{0+1}=7$