Is there a lens like this for the nikon d5100?

Is there a lens that has a very wide aperture but can still shoot to 300mm and have great quality images that are very sharp. So is there a lens for the nikon d5100 that is very wide but also is very long? I have looked but can't seem to find any that are not over 800 dollars.

Added (1). I would be using it for regular photos but also shooting wildlife from afar. I would like a lens that could take a photo of a flying bird and not be blurred and could also take family pictures. I would also need a lens for sporting events. Is there a lens like this? How much? Are there any cheap ones?

Check ebay every so often.

For the wide aperture there are several 300mm lenses available for the D5100 starting somewhere around $1500.
Or you can get a 70-200mm f/2.8 )OS by sigma for about $1300. OS is the same as VR.
Without the OS you can get the sigma 70-200 somewhere around $900
Or the Tamron 70-200 f/2.8 which is an awesome option if you are shooting anything but sports because the focus is pretty slow on it. It's a macro lens and they are slower to focus. It will shoot sports and do it fine, but the lack of fast focus will be felt by a sports shooter. That one runs about $700 I believe
I have the Sigma 70-200 OS and the Tamron 70-200 if you would like to see images.

Beyond that price there isn't a wide aperture lens that reaches that far. That wide, constant aperture is expensive to make.

You won't find an 18x, 24x, or any other super zooms in a DSLR like you can with a bridge camera. The reason for this is that those lenses are really not very good. Most super zooms have optical deficiencies, and are not consistently sharp throughout their focal length.

No one owning a DSLR would even buy such a high power zoom as they are really not that good.

What you want is a wide angle lens, a normal lens, and a telephoto lens, in which you change out as your needs dictate. That is the whole point of DLRS really, to have the ability to use different focal length lenses without the compromise in lens sharpness that bridge and compact cameras have.

http://www.althephoto.com

With telephoto lenses, 200mm and up, the widest aperture you can get is f/2.8. You can look at the Nikon/Nikkor 300mm f/2.8.It's great for wildlife and sports. For family pictures, stick to the 18-55mm kit lens for its wider angle of view. You can't get wide apertures with zoom lenses. That's the trade-off. Blame it to science.

Just for the laugh of it Sigma 200-500mm f/2.8

This ain't cheap you should follow the other answers!