As fringes are observed due to diffraction at two sources and diffraction has a particular area to cover. So we should be getting fringes only upto certain distance?
Diffraction can cover the whole screen left to right, theoretically out to 90 degrees. With a wide slit most of the light is concentrated more at the center, in contrast to the outer fringes. There are fringes out there but our eyes are more sensitive to contrast then absolute luminance. With a narrow slit the light is distributed more evenly across the screen leading to less contrast, so you can then notice the fringes way out there.
This is a good example of the difference between what you think that you might see from the theory and what is actually seen when doing the experiment.
The fringe pattern, after monochromatic light has passed through two slits, is a series of interference fringes whose separation is controlled by the slit separation with an intensity modulated by a single slit diffraction pattern whose width is controlled by the width of the slits.
In theory the region of interference is the whole of the "area" which is after the plane defined by the slits and remote from the source of light.
Thus, interference fringes should be produced within all that “area”.
However, there are two competing factors which usually make it impossible to observe all the fringes predicted from theory.
Almost all of the light is concentrated within the central maximum of the diffraction pattern with less and less light being concentrated within successive secondary maxima.
Making the central maximum wider by reducing the width of the slits produces a more uniform distribution of light across a greater "area" so more interference fringes will be produced within the central maximum of the diffraction envelope.
This increases the potential of observing more interference fringes not only because there are more fringes within the central maximum but also because those fringes are of similar brightness which means that the light from the centre of the pattern will not swamp the light from the pattern further away from the centre.
The downside is that reducing the width of the slits reduces the amount of light passing through the slits and this in turn reduces the brightness of the fringes making them harder to see.
So, with light passing through a double slit a compromise is often made as to the width of the slits by adjusting the slit width so that there is a comparatively narrow central maximum thus making the interference fringes bright enough to be easily seen but only over an "area" near near the central axis.
When using other types of waves, eg water waves in a ripple tank and microwaves, it is much easier to observe the interference patter over a much larger "area".