Answer
$ a.\quad$
Converges to $L=1.$.
See image. Steps given below.
$ b.\quad$
$N=5$
$N=41$
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
$a(x)=x\displaystyle \sin(\frac{1}{x})$
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 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$
$\epsilon=0.01.$
Define a function
$g(x)=|L-a(x)|$
We want to find an integer after which $ g(x)\lt 0.01.$
Looking at the table of values we find that $a(5)$ is within $0.01$ of L.
We take $N=5.$
$\epsilon=0.0001.$
Define a function
$g(x)=|L-a(x)|$
We want to find an integer after which $ g(x)\lt 0.0001.$
Looking at the table of values (for n ranging from 1 to 25), there are none.
We delete the table and create a new one, ranging from 26 to 50, step 1.
We find that $a(41)$ is within $0.0001$ of L.
We take $N=41.$
