A broadband connection to the internet has come to refer to a connection with a lot of bandwidth, or the ability to transfer large amounts of data in a short period of time.
Modem connections to the internet are low bandwidth, as anyone with a 56k modem trying to watch a streaming video over the internet knows! A 56k modem is capable of downloads of 5 kilobytes per second, under optimal conditions.
DSL and cable modem connections are common broadband connections, with typical download speeds ranging from 35 kilobytes per second to 75 kilobytes per second. This level of bandwidth is sufficent to enjoy streaming video.
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