alt Hacker NewsSchool calculus is hated because it's typically taught with epsilon delta proofs which is a formalism that happened later in the history of calculus. It's not that intuitive for beginners, especially students who haven't learn any logic to grok existential/universal quantifiers. Historically, mathematics is usually developed by people with little care for complete rigor, then they erase their tracks to make it look pristine. It's no wonder students are like "who the hell came up with all this". Mathematics definitely has an education problem.
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IMO, the calculus is taught incorrectly. It should start with functions and completely avoid sequences initially. Once you understand how calculus exploits continuity (and sometimes smoothness), it becomes almost intuitive. That's also how it was historically developed, until Weierstrass invented his monster function and forced a bit more rigor.
But instead calculus is taught from fundamentals, building up from sequences. And a lot of complexity and hate comes from all those "technical" theorems that you need to make that jump from sequences to functions. E.g. things like "you can pick a converging subsequence from any bounded sequence".
You can do it with infinitesimals if you like, but the required course in nonstandard analysis to justify it is a bastard.