The concept that makes calculus possible — understanding what happens as we approach, not just arrive.
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Direct substitution gives 0/0 (indeterminate!)
Factor: (x−2)(x+2)/(x−2) = x + 2 (for x ≠ 2)
lim (x→2) (x + 2) = 4
The function isn't defined at x = 2, but the limit exists. This is the key insight of calculus.
If f is continuous at c (no holes, jumps, or asymptotes), then lim(x→c) f(x) = f(c). All polynomials are continuous everywhere, so you can always plug in directly.
Factor, cancel, rationalize — use factoring techniques to eliminate the 0/0 form.
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This cannot be proved by algebra alone — it requires the Squeeze Theorem with geometric arguments from the unit circle. This limit is the foundation of all of trigonometric calculus.
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The two-sided limit exists only if both one-sided limits exist and are equal. One-sided limits are essential for understanding piecewise functions and step functions in probability.
What happens to f(x) as x → ∞? This determines the end behavior of functions and the existence of horizontal asymptotes.
For rational functions P(x)/Q(x), compare the degrees of P and Q — a technique from polynomial analysis. This idea extends to improper integrals and probability distributions.
A function f is continuous at x = c if three conditions hold:
If f is continuous on [a, b] and N is between f(a) and f(b), then there exists some c in (a, b) where f(c) = N. This guarantees that equations have solutions and is used in numerical methods for differential equations.
The rigorous definition of a limit, formalized by Weierstrass in the 19th century:
In plain English: no matter how small a tolerance ε you demand for the output, I can find a tolerance δ for the input that guarantees the output is within ε of L.