Answer
$ a.\quad$
Converges to $L=0.$.
See image. Steps given below.
$ b.\quad$
$N=650$ (is a possible answer)
$N=120,000$ (is a possible answer)
Work Step by Step
$a.\quad $
The steps you take will depend on the CAS you are using, but they follow the same logic.
Using the free online CAS at geogebra.org/cas:
Cell 1: Enter the function representing the sequence
$a(x)=\displaystyle \frac{\ln(x)}{x}$
From the dropdown menu, select "Table of values".
In the dialog box for the table, set the range from 1 to 25, step 1.
When we observe the graph, the points approach the x-axis (the line y=0).
The sequence seems to converge to 0.
In the next free cell of the CAS, we find the limit when $ n\rightarrow\infty$
Here, we enter "L=Limit(a, infinity)" (without quotes)
The CAS returns the limit to be $L=0$.
$ b.\quad$
$\epsilon=0.01.$
Changing the view window to $[500,1500]$ by $[-0.02,0.02]$
we graph the line $y=0.01$
and note that after $N=648$ we can safely say that $a_{n}$ is within $0.01$ of the limit L.
We take $N=650$.
$\epsilon=0.0001.$
Changing the view window to $[80000,500000]$ by $[-0.0002,0.0002]$
we graph the lines $y=\pm 0.0001$
and note that after $N=120,000$ we can safely say that $a_{n}$ is within $0.0001$ of the limit L.
We take $N=120,000$.
