The limit as x approaches a is basically calculus flexing its "I do what I want" energy. While mere mortals worry about what happens exactly AT a point, limits are out here living their best life, caring only about the neighborhood vibes around that point. The function could be completely undefined at a, could explode into infinity, could be taking a coffee break—doesn't matter. The limit just cruises by, checks out what's happening in the surrounding area, and calls it a day. It's like being invited to a party but only caring about what people are saying about it on the street outside. Pure mathematical confidence right there.
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