\[ \begin{split} \lim_{n \to \infty} \frac{7^n+5^n}{5^n+3^n}&=\lim_{n \to \infty} \frac{7^n\left(1+\left(\dfrac{5}{7}\right)^n\right)}{5^n\left(1+\left(\dfrac{3}{5}\right)^n\right)}=\\[16pt] &=\lim_{n \to \infty} \left(\frac{7}{5}\right)^n=\infty \end{split} \]