It looks so terrible that there is a huge temptation to use home cleaning products to remove it. However, resist that urge, as even the smallest residue can be lethal to fish. There are products, such as Safe Aquarium Cleaner, that are made to remove lime buildup on aquarium glass.
However, there is a readily available, inexpensive alternative – vinegar. That’s right, plain old vinegar. Lay the tank down on a towel, and pour enough vinegar on the affected glass to cover it. Leave it sit for 10-20 minutes, then scrub with a non-abrasive pad or cloth and rinse well. You may have to repeat the process a couple of times for each side to get it all off.