$\begin{array}{l}1)\quad \lim\dfrac{3^n -2.5^{n+1}}{2^{n+1} + 5^n}\\ = \lim\dfrac{3^n - 10.5^n}{2.2^n + 5^n}\\ = \lim\dfrac{\left(\dfrac35\right)^n -10}{2.\left(\dfrac26\right)^n +1}\\ =\lim\dfrac{0 - 10}{2.0 +1}\\ = -10\\ 2)\quad \lim\dfrac{2^n + 2^{n+1}}{2^n + 4.3^n}\\ =\lim\dfrac{3.2^n}{2^n + 4.3^n}\\ = \lim\dfrac{3.\left(\dfrac23\right)^n }{\left(\dfrac23\right)^n +4}\\ = \dfrac{3.0}{0+4}\\ =0\\ 3)\quad \lim\dfrac{4.3^n + 7^{n+1}}{7^n + 2.5^n}\\ = \lim\dfrac{4.3^n + 7.7^n}{7^n + 2.5^n}\\ = \lim\dfrac{4.\left(\dfrac37\right)^n +7}{1+2.\left(\dfrac57\right)^n }\\ = \dfrac{4.0 +7}{1 + 2.0}\\ = 7 \end{array}$
