$\begin{array}{l}4)\quad \lim\dfrac{1+3^n}{4+3^n}\\ =\lim\dfrac{\left(\dfrac13\right)^n +1}{4.\left(\dfrac13\right)^n +1}\\ = \dfrac{0+1}{4.0+1}\\ = 1\\ 5)\quad \lim\dfrac{4.3^n + 7^{n+1}}{2.5^n + 7^n}\\ = \lim\dfrac{4.3^n + 7.7^n}{2.5^n + 7^n}\\ = \lim\dfrac{4.\left(\dfrac37\right)^n +7}{2.\left(\dfrac57\right)^n + 1}\\ = \dfrac{4.0 +7}{2.0+1}\\ = 7\\ 6)\quad \lim\dfrac{4^{n+1} - 6^{n+2}}{5^n + 8^n}\\ = \lim\dfrac{4.4^n - 36.6^n}{5^n + 8^n}\\ = \lim\dfrac{4.\left(\dfrac48\right)^n - 26.\left(\dfrac68\right)^n }{\left(\dfrac58\right)^n + 1}\\ = \dfrac{4.0-36.0}{0+1}\\ =0 \end{array}$
