Answer
(a) The smallest possible value of $N$ is $N=3$.
(b) The smallest possible value of $N$ is $N=11$.
(c) The smallest possible value of $N$ is $N=1001$.
Work Step by Step
We have $\left| {{a_n} - L} \right| = \left| {\dfrac{1}{n} - 0} \right| \lt \varepsilon $ when $n \ge N$.
(a) For $\varepsilon = 0.5$,
$\dfrac{1}{n} \lt 0.5$
$n \gt 2$
The smallest possible value of $N$ is $N=3$.
(b) For $\varepsilon = 0.1$,
$\dfrac{1}{n} \lt 0.1$
$n \gt 10$
The smallest possible value of $N$ is $N=11$.
(c) For $\varepsilon = 0.001$,
$\dfrac{1}{n} \lt 0.001$
$n \gt 1000$
The smallest possible value of $N$ is $N=1001$.
