What to do when an external display isn't showing full resolution in OS X
There are several reasons why a monitor may not show output at its native resolution.
If you have attached a secondary display to your Mac, be it a computer monitor, a projector, or a television, you might naturally wish for the Mac to output video in the display's native resolution. This will keep the pixels at an even 1:1 ratio, resulting in a crisp image instead of one that blends pixels together, aliasing the image and making it look a touch fuzzy.
Unfortunately, sometimes when you connect an external display, the system will not output to its native resolution, and may show a blurred image, or show an image at a different aspect ratio or size, with black bars of unused screen real estate around the image.
This can be a frustrating issue to deal with if it happens, but in many cases it can be fixed.
First, be sure to check your Mac's display settings, and turn off mirroring modes. These modes allow the same image to be displayed on both monitors simultaneously, but in doing so the system will not be able to scale one resolution to match another, so it simply reduces the resolution of both to the highest shared resolution setting. This will result in one of the displays being blurry while the other is crisp.
Next, try using a different adapter or display cable, especially if the display is a large-format display. For instance, some monitors require a dual-link DVI connection, which means that many common single-link DVI adapters and cables will not work. While the display might still function at lower resolution outputs, this mode will result in the unwanted blurry image display.
If your monitor or television uses HDMI connections, you might find that in comparison to DVI or VGA connections, the system will often output an image that is less than the size of your monitor, resulting in a black border around the image. This can be addressed by dragging the Overscan control slider in the Displays system preferences until the image size matches the display. Unfortunately this can still result in the image appearing fuzzy unless you align the image perfectly, so if your external display supports DVI, VGA, or DisplayPort as input options, try using those as options.