Suppose you are travelling at a uniform velocity and you cover 1 meter in 1 second. Your average velocity is
$$\frac{1\ {\rm m}}{1\ {\rm s}} = 1 \frac{\rm m}{\rm s}.$$
If you consider a 1 millisecond interval within that 1 second, you cover 1 millimeter. Your average velocity in that milisecond is
$$\frac{1\ {\rm mm}}{1\ {\rm ms}} = 1 \frac{\rm m}{\rm s}.$$
If you consider a 1 microsecond interval within that 1 second, you cover 1 micron. Your average velocity in that microsecond is
$$\frac{1\ {\rm \mu m}}{1\ {\rm \mu s}} = 1 \frac{\rm m}{\rm s}.$$
If you consider a 1 nanosecond interval within that 1 second, you cover 1 nanometer. Your average velocity in that nanosecond is
$$\frac{1\ {\rm nm}}{1\ {\rm ns}} = 1 \frac{\rm m}{\rm s}.$$
No matter how small a time interval you consider, the distance traveled is proportionally reduced, and the average velocity covered in that time remains 1 m/s, rather than falling to 0.
Therefore we say that the limit of the average velocity, as the time interval approaches zero, is a non-zero value (1 m/s in my example).