2.5 The Precise Definition of a Limit

Limits are the foundation of calculus, describing how functions behave as they approach specific points. The epsilon-delta definition provides a precise way to understand this concept, using small intervals to capture the function's behavior near a given value.

This rigorous approach allows mathematicians to prove important properties of limits and develop key calculus concepts. By understanding the epsilon-delta definition, you'll gain a deeper insight into the fundamental ideas that drive calculus and mathematical analysis.

The Precise Definition of a Limit

Epsilon-delta definition of limits

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• Rigorous mathematical definition describing behavior of a function near a particular point
• For function $f(x)$ and limit $L$, limit of $f(x)$ as $x$ approaches $a$ is $L$ if for every positive number $\epsilon$ there exists a positive number $\delta$ such that whenever distance between $x$ and $a$ is less than $\delta$, distance between $f(x)$ and $L$ is less than $\epsilon$
• Symbolically: $\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0$ such that $0 < |x - a| < \delta \implies |f(x) - L| < \epsilon$
• Provides solid foundation for concept of limits in calculus
• Allows for precise statements about behavior of functions near a point crucial for developing concepts of continuity, derivatives, and integrals
• Eliminates ambiguity and allows for rigorous proofs of limit properties and theorems
• Uses absolute value to measure distances between points

Application of epsilon-delta approach

• To determine a function limit using epsilon-delta approach:

1. Choose an arbitrary positive value for $\epsilon$
2. Find a suitable value for $\delta$ in terms of $\epsilon$ such that whenever $0 < |x - a| < \delta$, it follows that $|f(x) - L| < \epsilon$
3. Prove that the chosen $\delta$ satisfies the condition for all $x$ within the delta-neighborhood of $a$, except possibly at $a$ itself

• Example: Prove $\lim_{x \to 2} (3x - 1) = 5$
  • Let $\epsilon > 0$ be given
  • $|f(x) - L| = |(3x - 1) - 5| = |3x - 6| = 3|x - 2|$
  • Choose $\delta = \frac{\epsilon}{3}$, so if $0 < |x - 2| < \delta$, then $|f(x) - L| = 3|x - 2| < 3 \cdot \frac{\epsilon}{3} = \epsilon$
  • Thus, for every $\epsilon > 0$, there exists a $\delta = \frac{\epsilon}{3} > 0$ such that $0 < |x - 2| < \delta \implies |(3x - 1) - 5| < \epsilon$, proving the limit

One-sided vs infinite limits

• One-sided limits approach a point from either left or right side
  • Left-hand limit: $\lim_{x \to a^-} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0$ such that $a - \delta < x < a \implies |f(x) - L| < \epsilon$
  • Right-hand limit: $\lim_{x \to a^+} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0$ such that $a < x < a + \delta \implies |f(x) - L| < \epsilon$
  • For a limit to exist, both left-hand and right-hand limits must exist and be equal
• Infinite limits occur when function values become arbitrarily large or small as $x$ approaches a certain point
  • Limit approaching positive infinity: $\lim_{x \to a} f(x) = \infty \iff \forall M > 0, \exists \delta > 0$ such that $0 < |x - a| < \delta \implies f(x) > M$
  • Limit approaching negative infinity: $\lim_{x \to a} f(x) = -\infty \iff \forall M < 0, \exists \delta > 0$ such that $0 < |x - a| < \delta \implies f(x) < M$
• Both one-sided and infinite limits can be defined using epsilon-delta definition with slight modifications to inequalities and interpretation of limit value

Epsilon-delta support for limit laws

• Epsilon-delta definition of limits provides rigorous foundation for limit laws used to evaluate limits of combinations of functions
• Examples of limit laws supported by epsilon-delta definition:
  • Sum rule: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
    • Proof: Let $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$. For every $\epsilon > 0$, there exist $\delta_1, \delta_2 > 0$ such that $|f(x) - L| < \frac{\epsilon}{2}$ when $0 < |x - a| < \delta_1$ and $|g(x) - M| < \frac{\epsilon}{2}$ when $0 < |x - a| < \delta_2$. Choose $\delta = \min(\delta_1, \delta_2)$, then $|[f(x) + g(x)] - (L + M)| \leq |f(x) - L| + |g(x) - M| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$ when $0 < |x - a| < \delta$
  • Constant multiple rule: $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$, where $c$ is a constant
    • Proof: Let $\lim_{x \to a} f(x) = L$. For every $\epsilon > 0$, there exists a $\delta > 0$ such that $|f(x) - L| < \frac{\epsilon}{|c|}$ when $0 < |x - a| < \delta$. Then, $|c \cdot f(x) - c \cdot L| = |c| \cdot |f(x) - L| < |c| \cdot \frac{\epsilon}{|c|} = \epsilon$ when $0 < |x - a| < \delta$
• Proofs demonstrate how epsilon-delta definition supports limit laws, ensuring their validity and consistency within framework of calculus

Mathematical Foundations

• Functions: Mappings between sets of numbers, typically from real numbers to real numbers in calculus
• Real numbers: The set of all rational and irrational numbers, represented by points on a number line
• Inequalities: Mathematical statements expressing the relative size or order of two values, crucial in defining neighborhoods and intervals in limit definitions