In case anyone else finds this thread and is appalled by a $400-$500 quote from the mechanic, the cheapest place to find a replacement mirror for a 2013 Elantra is partsgeek. $75 as of today for a heated mirror with turn signal, $50 without. Both will fit any 2013 Elantra.
A direct link:
The only problem is, it only comes in BLACK. If you're like me and your car isn't black, you can either spray paint the mirror cover, or you can just buy another unpainted mirror cover for your other side you aren't replacing so they look the same.
I found the mirror covers on ebay, just search for "2013 Elantra mirror cover", currently you can find covers with and without the turn signal, and in chrome and black. They are $15 with free shipping.
If you don't want both your mirrors to be black, you can buy a matched color can, one site is automotive touchup. They will guarentee a color match for any car. You will need to give them your cars paint information. On an Elantra it's in the driver-side door jam (where the door lock mechanism meets the car frame).
Here is a direct link for a paint-matched can:
Its $20.
All together this cost me less than $100 and about 30 minutes including research and replacement time, the mechanic I went to quoted me $485. I feel disgusted and enlightened at the same time.
To replace the mirror assembly, you just need 3 things:
1) A thin plastic shim (or flat blade screwdriver with a thin cloth to protect your plastic) to remove the door mirror panel.
2) A 10mm driver to remove 3 bolts holding the mirror assembly to the car frame.
3) A phillips driver to remove another screw.
You can watch this video which shows you everything (it's for an older model Elantra, but the process is exactly the same)
Boom! Done! Now you can buy a PS4 with the money you saved! [BLEEP] Yeah! You're awesome!!