Answer
$ a.\quad$
Converges to $L=0.$.
See image. Steps given below.
$ b.\quad$
$N=46,050$ (is a possible answer)
$N=92,100$ (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
$a(x)=(0.9999)^x$
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 seem to drop away from the line $y=1.$
Zooming out sufficiently, we see that they 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$
$\epsilon=0.01.$
Keeping the zoom as above, we find where the graph of $a(x)$ intersects the line $y=0.01$
The cas returns $x=46,049.399$
So, we can take $N=46,050$.
$\epsilon=0.0001.$
Keeping the zoom as above, we find where the graph of $a(x)$ intersects the line $y=0.0001$
The cas returns $x=92098.798$
So, we can take $N=92,100$
