\(\lim[\sqrt{n+1}(\sqrt{n+3}-\sqrt{n+2})] =\lim[\sqrt{n+1}(\dfrac{n+3-n-2}{\sqrt{n+3}+\sqrt{n+2}})] =\lim[\sqrt{n}.\sqrt{1+\dfrac{1}{n}}.\dfrac{1}{\sqrt{n}.\sqrt{1+\dfrac{3}{n}}+\sqrt{n}.\sqrt{1+\dfrac{2}{n}}}] =\dfrac{1}{2}\)

\(\lim\dfrac{1+2.3^{n}-7^{n}}{a+5^{n}+a.7^{n-1}} =\lim\dfrac{(\dfrac{1}{7})^{n}+2.(\dfrac{3}{7})^{n}-1}{a.(\dfrac{1}{7})^{n}+(\dfrac{5}{7})^{n}+\dfrac{a}{7}} =\lim\dfrac{-1}{\dfrac{a}{7}} =\dfrac{-7}{a}\)

\(\lim\dfrac{2^{n}(4^{n+1}-3^{n+2}-1)}{5^{n}+8^{n}} =\lim\dfrac{4.8^{n}-9.6^{n}-2^{n}}{5^{n}+8^{n}} =\lim\dfrac{4-9.(\dfrac{6}{8})^{n}-(\dfrac{2}{8})^{n}}{(\dfrac{5}{8})^{n}+1} =\lim\dfrac{4-9.0-0}{0+1} =4\)