Unit 1 builds the foundation of calculus — limits describe how functions behave at and near specific values. Continuity and the Intermediate Value Theorem follow from a solid understanding of limits.
Limit: the value f(x) approaches as x approaches c. Direct substitution: try first. If 0/0 (indeterminate): factor, rationalise (conjugate), or use special limits (limₓ→₀ sinx/x = 1). One-sided limits: lim from left (x→c⁻) and right (x→c⁺). Limit exists only if both one-sided limits are equal. Squeeze theorem: if g(x) ≤ f(x) ≤ h(x) and limₓ→c g(x) = limₓ→c h(x) = L, then limₓ→c f(x) = L.
f is continuous at x = c if: (1) f(c) exists, (2) lim_{x→c} f(x) exists, (3) lim_{x→c} f(x) = f(c). Types of discontinuity: removable (hole), jump, infinite (vertical asymptote). IVT: if f is continuous on [a,b] and N is between f(a) and f(b), then there exists c in (a,b) with f(c) = N. Application: prove existence of roots.
Horizontal asymptote: lim_{x→∞} f(x) = L. For rational functions: compare degrees. Degree top < degree bottom → y = 0. Equal degrees → y = leading coefficients ratio. Top degree > bottom → no horizontal asymptote. Infinite limits: vertical asymptotes where denominator → 0 and numerator doesn\'t.
The form 0/0 is "indeterminate" because it does not determine the limit\'s value — the actual limit could be any real number, ∞, or not exist, depending on the functions involved. For example: lim_{x→0} x/x = 1, lim_{x→0} x²/x = 0, lim_{x→0} x/x² = ∞ — all are 0/0 forms with different answers. This is why you must use algebraic techniques (factoring, L\'Hôpital\'s rule) to simplify the expression and find the actual limit. Other indeterminate forms include ∞/∞, 0·∞, ∞-∞, 0⁰, 1^∞, ∞⁰.
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