[tex]\lim_{n \to \infty} \left(\frac{5*7^{2n+1}-7^n+7}{-5*49^n-2}\right) = \lim_{n \to \infty} \left(\frac{5*7*7^{2n}-7^n+7}{-5*7^{2n}-2}\right)= \lim_{n \to \infty} \frac{7^{2n}(35-\frac{7^n}{7^{2n}}+\frac{7}{7^{2n}})}{7^{2n}(-5-\frac{2}{7^{2n}})} =\lim_{n \to \infty} \frac{35-\frac{1}{7^n}+\frac{1}{7^{2n-1}}}{-5-\frac{2}{7^{2n}}} =[\frac{35}{-5}]=-7[/tex]
[tex]f'(x)=(\frac{x+2}{x-2})'=\frac{(x+2)'(x-2)-(x+2)(x-2)'}{(x-2)^2}=\frac{x-2-x-2}{(x-2)^2}=\frac{-4}{(x-2)^2}[/tex]
[tex]\lim_{n \to \infty} (\sqrt{3n^2-6n+4}-\sqrt3n)= \lim_{n \to \infty} (\sqrt{3n^2-6n+4}-\sqrt3n)*\frac{\sqrt{3n^2-6n+4}+\sqrt3n}{\sqrt{3n^2-6n+4}+\sqrt3n}=\\\lim_{n \to \infty}\frac{3n^2-6n+4+3n^2}{\sqrt{3n^2-6n+4}+\sqrt3n}=\lim_{n \to \infty}\frac{-6n+4}{\sqrt{3n^2-6n+4}+\sqrt3n}=\lim_{n \to \infty}\frac{n(-6+\frac{4}{n})}{n\left(\sqrt{3-\frac{6}{n}+\frac{4}{n^2}}+\sqrt3\right)}=[/tex][tex]=\lim_{n \to \infty}\frac{-6+\frac{4}{n}}{\sqrt{3-\frac{6}{n}+\frac{4}{n^2}}+\sqrt3}=\frac{-6}{2\sqrt3}=\frac{-3}{\sqrt3}*\frac{\sqrt3}{\sqrt3}=\frac{-3\sqrt3}{3}=-\sqrt3[/tex]
[tex]\lim_{n \to \infty} \left(\frac{4n^3+5n^2-6n}{-2n^3-4n^2+1}\right)= \lim_{n \to \infty} \frac{n^3(4+\frac{5}{n}-\frac{6}{n^2})}{n^3(-2-\frac{4}{n}+\frac{1}{n^2})}=\lim_{n \to \infty} \frac{4+\frac{5}{n}-\frac{6}{n^2}}{-2-\frac{4}{n}+\frac{1}{n^2}}=-2[/tex]
