Best Zoom Lens for Sporting Events?

We're getting a camera for my mom for Christmas and she wants to be able to take pictures from stands and zoom into a players face or just get a picture close to the players. If it helps, we're getting the Nikon D3100.

Go for a 400-500mm lens. They usually cost $600+

Nikon 70-300mm or 55-200mm

I would consider Nikon 18-200mm f/3.5-5.6G AF-S ED VR II Nikkor Telephoto Zoom Lens for Nikon DX-Format Digital SLR Cameras

One-lens solution adept in a wide variety of situations
Nikon VR II (Vibration Reduction) image stabilization
Two Extra-low Dispersion (ED) elements; three aspherical lens elements
Exclusive Nikon Silent Wave Motor (SWM)
Focus to 20 inches for extended versatility
Nikon Super Integrated Coating (SIC)