Depends what “Went out” means exactly. You can do thread search for "Outside Temperature Display", "Ambient Temperature Display", or either with "Sensor" added and you should find a couple of threads that detail how to diagnose and repair problems with the sensor. The sensor is usually located under the front bumper somewhere. This would be the first step unless you can see a defective LCD panel.

This system can be divided into three basic system blocks. The sensor and its' wire leads, the electronic digital circuitry, and the LCD panel itself. The electronics and LCD are contained in one unit attached to the underside of the instrument cluster. You pull the instrument cluster and remove two screws, then pull the Outside Temperature Display unit free. You unplug a four conductor connector that provides the unit power, ground, illumination, and sensor connections.

If there is nothing occurring on the instrument cluster window then any one of the three parts could be at fault.

If you see anything on the display such as black blob appearance on all or a portion it may be the Liquid Crystal Display panel itself is physically damaged, cracked, or broken. The only solution is to replace the panel itself. The only place I have heard off to obtain the panel is another display unit from a salvage yard or the like, you wouldn't need working electronics just an intact LCD panel. You can disassemble the unit and carefully lift out your damaged panel and replace with an undamaged panel. Like wise if your electronics have failed you can find a unit with a damaged LCD panel and swap the parts up and put together the functioning pieces to create a functioning whole.

If you see partial numbers or intermittent blinking numeral segments it could be the electronics or the connections to the LCD panel. You could disassemble and look around or swap out known parts.

Do you have a C or F display? If what was said earlier is true, a C will be easier to find. The only place I’ve heard of for a new replacement unit is MBUSA at around $175! So it could pay to do your homework and ascertain exactly what went wrong. Good luck!
The F part number is 126 542 0227.