Answer
(a) The smallest possible value of $N$ is $N=4$.
(b) The smallest possible value of $N$ is $N=10$.
(c) The smallest possible value of $N$ is $N=1000$.
Work Step by Step
We have $\left| {{a_n} - L} \right| = \left| {\dfrac{n}{{n + 1}} - 1} \right| \lt \varepsilon $, when $n \ge N$.
(a) For $\varepsilon = 0.25$,
$\left| {\dfrac{n}{{n + 1}} - 1} \right| \lt 0.25$
$\left| {\dfrac{{ - 1}}{{n + 1}}} \right| \lt 0.25$
$\dfrac{1}{{n + 1}} \lt 0.25$
$n \gt 3$
The smallest possible value of $N$ is $N=4$.
(b) For $\varepsilon = 0.1$,
$\left| {\dfrac{n}{{n + 1}} - 1} \right| \lt 0.1$
$\left| {\dfrac{{ - 1}}{{n + 1}}} \right| \lt 0.1$
$\dfrac{1}{{n + 1}} \lt 0.1$
$n \gt 9$
The smallest possible value of $N$ is $N=10$.
(c) For $\varepsilon = 0.001$,
$\left| {\dfrac{n}{{n + 1}} - 1} \right| \lt 0.001$
$\left| {\dfrac{{ - 1}}{{n + 1}}} \right| \lt 0.001$
$\dfrac{1}{{n + 1}} \lt 0.001$
$n \gt 999$
The smallest possible value of $N$ is $N=1000$.
