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I think this is probably quite dependent on what’s normal for ISPs in the region. In the UK for example, every ISP router I’ve had runs a DNS server and it’s that which is given out via DHCP. It then forwards onto the ISPs DNS platform.
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In Scotland I was with Telewest, then Virgin, and my memory is always that the DHCP pushed out the external IP of the ISP's DNS servers.

Nowadays I'm in Finland and definitely the router runs no DNS service, the DHCP service advertises the ISP resolvers.

Probably depends on the region/ISP I guess, but I had no expectation that it would be the more common option.

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American here, most of ISPs here do it as well. With modern router hardware, there is plenty of hardware available to run tiny DNS server that caches and forwards all requests to ISP upstream. Memory overhead is probably about 50MB and CPU overhead is trivial, probably .1% or less.
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My parents are with Bell (the biggest ISP in Canada) and use the Bell Gigahub (Router/AP/Switch in one). It does have a DNS cache and the its set as the DNS resolver in their DHCP configuration.
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The system's configured DNS resolver is usually your router.
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I would argue the contrary - most home routers are running a DNS server of some kind. They forward to upstream, but will resolve local names like your printer and whatnot.

dnsmasq is the defacto tool on these embedded devices for dhcp+dns. probably a billion deployments. it's up there with sqlite for most used tech.

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IIRC a resolver is what people would think of as a DNS server only it's not an authority for any domains. Like you said, they're used to get load off of authoritative servers and are very common. I think dnsmasq is mentioned explicitly in the O'Reilly locust book but it's been a while.
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