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At the insulation around the wires - if it's dry, cracked, or comes off easily, that's a bad sign. Wires that get sticky or smell weird are also a sign that the rubber/plastic is breaking down. If you find spots where the copper is showing or if the wires are brittle, you really need to replace them before you have a problem. Ask an electrician to check if you're not sure, it's not expensive and it's worth it for safety.
First check the insulation around the wires - if it's cracked, hardened, or crumbling to the touch, that's a bad sign and you need to replace them. The color can tell you something too; if it's turned brown or black instead of its original color, that means it's aged. And if you've got old wiring that's more than 30-40 years old in your walls, honestly it's not a bad idea to have someone check it out before you end up with short circuits or worse.
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