a: \(\lim_{}\frac{4^{n-1}-3\cdot5^{n+1}}{25^{n}-3^{n-1}}\)
\(=\lim_{}\frac{4^{n}\cdot\frac14-15\cdot5^{n}}{25^{n}-3^{n}\cdot\frac13}\)
\(=\lim_{}\frac{\frac{4^{n}}{25^{n}}\cdot\frac14-15\cdot\frac{5^{n}}{25^{n}}}{\frac{25^{n}}{25^{n}}-\frac{3^{n}}{25^{n}}\cdot\frac13}=\lim_{}\frac{\left(\frac{4}{25}\right)^{n}\cdot\frac14-15\cdot\left(\frac{5}{25}\right)^{n}}{1-\left(\frac{3}{25}\right)^{n}\cdot\frac13}=0\)
b: \(\lim_{}\frac{7^{n-1}-3^{n+1}}{4^{n}-5^{n-1}}\)
\(=\lim_{}\frac{7^{n}\cdot\frac17-3^{n}\cdot3}{4^{n}-5^{n}\cdot\frac15}\)
\(=\lim_{}\frac{\frac{7^{n}}{7^{n}}\cdot\frac17-\frac{3^{n}}{7^{n}}\cdot3}{\frac{4^{n}}{7^{n}}-\frac{5^{n}}{7^{n}}\cdot\frac15}=-\) ∞
