Answer
$\displaystyle \lim_{n\rightarrow\infty}a_{n}=2$
Work Step by Step
Our inference, based on the graph (see below)
is that the sequence converges to 2.
Analytically, we use Th.9.1:
"Let $L$ be a real number.
Let $f$ be a function of a real variable such that $\displaystyle \lim_{x\rightarrow\infty}f(x)=L$.
If $\{a_{n}\}$ is a sequence such that $f(n)=a_{n}$ for every positive integer $n$,
then $\displaystyle \lim_{n\rightarrow\infty}a_{n}=L$. "
We define $f(x)=2-\displaystyle \frac{1}{4^{x}}$, for which $a_{n}=f(n)$ .
As $ x\rightarrow\infty$, the denominator becomes very large, and $f(x)\rightarrow 2-0=2.$
So,
$\displaystyle \lim_{n\rightarrow\infty}a_{n}=\lim_{x\rightarrow\infty}f(x)=2$.
