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@Alfie, the basic rule of thumb is, you never want to limit content based on your audience’s preferred device (don’t present content on the desktop that isn’t on the mobile), only change the presentation of the content, BUT, with that being said, there may be scenarios that you won’t want to present that blog post thumbnail to a mobile user, because of the precious bandwidth, that’s where scripts like response.js come in, where they can assist you with dynamically serving images/blocks of code, based on screen resolution.
I agree with @ChrisP — I believe content should always be the same regardless of the device that’s being used. And, images (static or slider) I would consider content too. Having said that, Chris’ solution for response.js would indeed do the trick.
In general however, if you have anything that should not be seen on small screens but should be seen on higher screen widths, the display:none method is the easiest, most straightforward way to do it.