The asymptote cannot be x= because x can be any number. If you think about it, you can take a number to any exponent.
If x is a positive exponent, y is positive.
If x is a nevative exponent, y decreases, but is still positive. This is because a number to a negative exponent equals 1 over the number to the positive exponent. Thus, it is smaller, but still positive.
If x is 0, y is positive again because anything to the zero is positive 1.
There is no way y could be less than or equal to zero. So, there is an asymptote at y=0.
Also, set the equation equal to 0 and solve. You should end up with 4^x=0. Since no exponenent can make a number zero, this isn't possible, so y cannot equal zero.
Here is the graph for a visual: