UHD 4k 64MP Video Camera Camcorder with 18X Digital Zoom, Kentfaith Video Camera 4.0" Rotating Touchscreen,Microphone,Remote Control, 64GB SD Card, Two Batteries

SKU: GW41.0091

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I haven’t used it for sports but it has a pretty fast zoom, so I would assume that itv would do just fine. 
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Camcorder Video Camera 4K supports external microphone with 3.5mm connector. 
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HDMI output can only play video and photos. The connection can be connected in any mode of the camera. 
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Yes, the video camera can record while it is charging. 
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Yes, it can work as a webcam as long as your device can connect to the camera. 
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It HAS an Auto Focus button. Works great! As for the Anti-vibration/shake, NO, not that I can see. However, a hand stabilizer handle should help with that. I haven't notice any vibrations although I'm still experimenting. Beautiful video! FYI,Can't find an external monitor that works with this camera. Purchased two different monitors and five different HDMI Cables, No luck. But the flip out built-in is fine. Good quality, I just wanted something larger.