Answer
$ a.\quad$
Converges to $L=1.$.
See image. Steps given below.
$ b.\quad$
$N=1180$
$N=117,250$
(possible answers)
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
$a(x)= (123456)^{1/x}$
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 towards the x-axis.
Zooming out sufficiently, we see that they approach the line $y=1$.
The sequence seems to converge to $1$.
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 $L=1$.
$ b.\quad$
$\epsilon=0.01.$
Zooming out, we find where the graph of $a(x)$ intersects the line $y=1.01$
The cas returns $x=1178.216$
So, we can take $N=1180$.
$\epsilon=0.0001.$
Keeping the zoom as above, we find where the graph of $a(x)$ intersects the line $y=1.0001$
The cas returns $x=117,242.26$
So, we can take $N=117,250$
