The concept that makes calculus possible — understanding what happens as we approach, not just arrive.
A limit describes the value that a function approaches as its input approaches a particular value. We don't care what happens at the point — only what happens near it.
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.
If lim f(x)/g(x) gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x). This requires derivatives, covered in the next lesson.
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.
Sometimes the limit depends on which direction you approach from:
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.