Question

Evaluate the limit :-

${ \sf { \lim_{ n \rightarrow \infty { \bigg ( \dfrac{n+2}{2n+7} } \bigg ) } } }$ ​

Answer 1

$\bold \red{\lim_{n \to \infty} (\frac{n + 2}{2 n + 2}})$

$\small{\bold \red{\lim_{n \to \infty} c f{\left(n \right)} = c \lim_{n \to \infty} f{\left(n \right)} \: with \: c=\frac{1}{2} \: and \: f{\left(n \right)} = \frac{n + 2}{n + 1}:}}$

$\color{red}{\lim_{n \to \infty} \frac{n + 2}{2 n + 2}} = \color{red}{\left(\frac{\lim_{n \to \infty} \frac{n + 2}{n + 1}}{2}\right)}$

$\bold{\frac{\color{green}{\lim_{n \to \infty} \frac{n + 2}{n + 1}}}{2} = \frac{\color{red}{\lim_{n \to \infty} \frac{n \frac{n + 2}{n}}{n \frac{n + 1}{n}}}}{2}}$

$\bold{ \pink{\frac{\color{red}{\lim_{n \to \infty} \frac{n \frac{n + 2}{n}}{n \frac{n + 1}{n}}}}{2} = \frac{\color{red}{\lim_{n \to \infty} \frac{1 + \frac{2}{n}}{1 + \frac{1}{n}}}}{2}}}$

$\bold{ \blue{\frac{\color{re}{\lim_{n \to \infty} \frac{1 + \frac{2}{n}}{1 + \frac{1}{n}}}}{2} = \frac{\color{red}{\frac{\lim_{n \to \infty}\left(1 + \frac{2}{n}\right)}{\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)}}}{2}}}$

$\green{\frac{\color{red}{\lim_{n \to \infty}\left(1 + \frac{2}{n}\right)}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)} = \frac{\color{red}{\left(\lim_{n \to \infty} 1 + \lim_{n \to \infty} \frac{2}{n}\right)}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)}}$

$\frac{\lim_{n \to \infty} \frac{2}{n} + \color{red}{\lim_{n \to \infty} 1}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)} = \frac{\lim_{n \to \infty} \frac{2}{n} + \color{red}{1}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)}$

${ \bold \green{\lim_{n \to \infty} \:c\: f{\left(n \right)} = c \lim_{n \to \infty} f{\left(n \right)} \:with \:c=2 \:and\: f{\left(n \right)} = \frac{1}{n}}}$

$\frac{1 + \color{red}{\lim_{n \to \infty} \frac{2}{n}}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)} = \frac{1 + \color{red}{\left(2 \lim_{n \to \infty} \frac{1}{n}\right)}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)}$

$\frac{1 + 2 \color{red}{\lim_{n \to \infty} \frac{1}{n}}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)} = \frac{1 + 2 \color{red}{\frac{\lim_{n \to \infty} 1}{\lim_{n \to \infty} n}}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)}$

$\red{\frac{1 + \frac{2 \color{red}{\lim_{n \to \infty} 1}}{\lim_{n \to \infty} n}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)} = \frac{1 + \frac{2 \color{red}{1}}{\lim_{n \to \infty} n}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)}}$

$\pink{\frac{1 + 2 \color{blue}{1 \frac{1}{\lim_{n \to \infty} n}}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)} = \frac{1 + 2 \color{red}{\left(0\right)}}{2 \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)}}$

${ \bold \green{\frac{\color{red}{\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)}^{-1}}{2} = \frac{\color{red}{\left(\lim_{n \to \infty} 1 + \lim_{n \to \infty} \frac{1}{n}\right)}^{-1}}{2}}}$

$\bold{\frac{\left(\lim_{n \to \infty} \frac{1}{n} + \color{red}{\lim_{n \to \infty} 1}\right)^{-1}}{2} = \frac{\left(\lim_{n \to \infty} \frac{1}{n} + \color{red}{1}\right)^{-1}}{2}}$

$\bold{\frac{\left(1 + \color{red}{\lim_{n \to \infty} \frac{1}{n}}\right)^{-1}}{2} = \frac{\left(1 + \color{red}{\frac{\lim_{n \to \infty} 1}{\lim_{n \to \infty} n}}\right)^{-1}}{2}}$

$\purple{\frac{\left(1 + \frac{\color{red}{\lim_{n \to \infty} 1}}{\lim_{n \to \infty} n}\right)^{-1}}{2} = \frac{\left(1 + \frac{\color{red}{1}}{\lim_{n \to \infty} n}\right)^{-1}}{2}}$

$\red{\frac{\left(1 + \color{red}{1 \frac{1}{\lim_{n \to \infty} n}}\right)^{-1}}{2} = \frac{\left(1 + \color{red}{\left(0\right)}\right)^{-1}}{2}}$

$\bold{\lim_{n \to \infty} (\frac{n + 2}{2 n + 2} = \frac{1}{2}})$