[tex]\displaystyle\\\lim_{n\to\infty}\left(\dfrac{n+7}{n+10}\right)^{4n+2}=\\\\\lim_{n\to\infty}\left(\left(\dfrac{n+10}{n+10}+\left(-\dfrac{3}{n+10}\right)\right)^{-\dfrac{n+10}{3}}\right)^{-\dfrac{3(4n+2)}{n+10}}=\\\\\lim_{n\to\infty}\left(\left(1+\left(-\dfrac{3}{n+10}\right)\right)^{-\dfrac{n+10}{3}}\right)^{-\dfrac{3(4n+2)}{n+10}}=\\\\e^{\displaystyle\lim_{n\to\infty}-\dfrac{3(4n+2)}{n+10}}=\\\\e^{\displaystyle\lim_{n\to\infty}-\dfrac{12n+6}{n+10}}=[/tex]
[tex]e^{\displaystyle\lim_{n\to\infty}-\dfrac{12n+120-114}{n+10}}=\\\\e^{\displaystyle\lim_{n\to\infty}-\dfrac{12(n+10)-114}{n+10}}=\\\\e^{\displaystyle\lim_{n\to\infty}\left(-12+\dfrac{114}{n+10}\right)}=\\\\e^{\displaystyle-12+\lim_{n\to\infty}\left(\dfrac{114}{n+10}\right)}=\\\\e^{-12+0}=e^{-12}=\dfrac{1}{e^{12}}[/tex]
