Chapter 3: Limits and Continuity
Introduction
Limits are the foundation of calculus. They describe behavior of functions as inputs approach specific values. Understanding limits is essential for defining continuity, derivatives, and integrals. This chapter introduces limit concepts, techniques for computing limits, and the notion of continuity.
1. Concept of a Limit
Informal Definition
We say "the limit of f(x) as x approaches a is L" (written lim(x→a) f(x) = L) if f(x) gets arbitrarily close to L as x gets arbitrarily close to a (but x ≠ a). The key: we approach a but never equal it.
Formal Definition (ε-δ)
For every ε > 0 (no matter how small), there exists δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This says f(x) stays within ε of L whenever x is within δ of a.
2. Techniques for Finding Limits
Direct Substitution
If f is continuous at a, then lim(x→a) f(x) = f(a). Simply substitute. Works for polynomials, trigonometric functions, and other continuous functions.
Factoring and Cancellation
For indeterminate forms like 0/0, factor the numerator and denominator. If a common factor appears, cancel it and then substitute. Example: lim(x→2) (x² - 4)/(x - 2) = lim(x→2) (x - 2)(x + 2)/(x - 2) = lim(x→2) (x + 2) = 4.
Rationalization
For limits involving square roots, multiply by the conjugate to eliminate the root. Example: lim(x→0) √(x + 1) - 1/x. Multiply by (√(x + 1) + 1)/(√(x + 1) + 1), simplify, then substitute.
Limits at Infinity
For lim(x→∞) f(x), examine the leading terms. For rational functions, the limit depends on the degree of numerator vs. denominator. If degrees are equal, the limit is the ratio of leading coefficients.
3. Limit Laws
Sum and Difference
If lim(x→a) f(x) = L and lim(x→a) g(x) = M, then lim(x→a) [f(x) ± g(x)] = L ± M.
Product and Quotient
lim(x→a) [f(x)g(x)] = LM and lim(x→a) [f(x)/g(x)] = L/M (provided M ≠ 0).
Power Rule
lim(x→a) [f(x)]^n = L^n for positive integer n.
4. Continuity
Definition of Continuity
A function f is continuous at point a if three conditions hold: (1) f(a) is defined, (2) lim(x→a) f(x) exists, (3) lim(x→a) f(x) = f(a). Intuitively: no jumps, holes, or breaks.
Types of Discontinuity
Removable discontinuity: limit exists but f(a) is undefined or wrong (hole in graph). Jump discontinuity: left and right limits exist but are unequal (jump in graph). Infinite discontinuity: limit is ∞ (vertical asymptote).
Continuity on Intervals
f is continuous on [a, b] if it's continuous at every point in the interval. Continuous functions on closed intervals have nice properties: they achieve maximum and minimum values (Extreme Value Theorem).
5. Important Theorems
Intermediate Value Theorem
If f is continuous on [a, b] and N is between f(a) and f(b), then there exists at least one c in (a, b) where f(c) = N. In simple terms: a continuous function must cross every value between its endpoints.
Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) and lim(x→a) g(x) = lim(x→a) h(x) = L, then lim(x→a) f(x) = L. The middle function is "squeezed" to the limit of the bounds.
Conclusion
Limits formalize the intuitive notion of approaching a value. They're essential for defining continuity and, later, derivatives. Mastering limit techniques and understanding continuity prepares you for the core of calculus.