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The limit of a function f(x) as x approaches a value c is the value that f(x) gets closer and closer to as x gets closer to c. We write this as lim(x→c) f(x) = L, meaning f(x) can be made arbitrarily close to L by choosing x sufficiently close to c.
For every ε > 0 there exists a δ > 0 such that if 0 < |x − c| < δ, then |f(x) − L| < ε. This formal definition precisely captures the idea that f(x) can be made as close to L as desired by restricting x to be close enough to c.
A function f is continuous at c if three conditions hold: (1) f(c) is defined, (2) lim(x→c) f(x) exists, and (3) lim(x→c) f(x) = f(c). Intuitively, you can draw the graph through c without lifting your pen.
If g(x) ≤ f(x) ≤ h(x) for all x near c (except possibly at c), and lim(x→c) g(x) = lim(x→c) h(x) = L, then lim(x→c) f(x) = L. This is useful for evaluating limits of functions trapped between two others that share the same limit.
A one-sided limit considers the behavior of f(x) as x approaches c from only one direction. lim(x→c⁺) f(x) is the right-hand limit and lim(x→c⁻) f(x) is the left-hand limit. The two-sided limit exists only if both one-sided limits exist and are equal.