Finding unknown values from discrete probabilities. 
(I am confused here with the limits. It says x = 0,1,2,3... So what is my end limit her? Thanks.)
 A: This probability density function has mass on $\mathbb{N}_0$ which is all natural numbers including $0$. So you're looking at a random variable having mass at infinitely (though countably) many points, and hence this is an exercise in evaluating a series. 
A: $\sum_{x=0}^\infty k(\frac{4}{5})^x = 1$
We are finding sum of a geometric series
A geometric series is of the form 
$S_n = a + ar^1 + ar^2 + ... + ar^n$ 
$S_nr - S_n = ar^{n+1}- a$
$S_n = a[\frac{r^{n+1}-1}{r - 1}]$
In this case our r =4/5
and a= k
$1 = \lim_{n->\infty}  k[\frac{(0.8^{n+1}) - 1}{0.8 - 1}]$
$1 = k \frac{0 - 1}{-0.2}$
$1 = k (5)$
$k = 1/5$

I am confused here with the limits. It says x = 0,1,2,3... So what is
  my end limit her? Thanks

Infinity, it keeps going
A: There is no end limit, it is just infinity... So, you have to sum up $g(x)$ from $x=0$ to $\infty$, and make it equal to $1$ to find $k$. 
This can be easily done, since the summation is a geometric series: 
$\sum_{x=0}^\infty k\left(\frac{4}{5}\right)^x = k \frac{1}{1-\frac{4}{5}} = 5k$.
So, $k$ should be equal to $\frac{1}{5}$.
