Overview

Limits are the foundation of calculus. This topic introduces how to evaluate the behavior of a function as it approaches a particular point and how limits help define continuity, derivatives, and integrals. Understanding limits is crucial for all later topics.

Key Concepts and Structures

• Definition of a Limit: \lim_{x \to a} f(x) = L means that as x approaches a, f(x) approaches L.
• Limit Laws: You can add, subtract, multiply, and divide limits (as long as denominators aren’t zero).
• One-sided Limits: Left-hand limit \lim_{x \to a^-} and right-hand limit \lim_{x \to a^+} must match for the full limit to exist.
• Limits at Infinity: What happens to f(x) as x becomes very large or very small. Rational functions approach horizontal asymptotes.
• Indeterminate Forms: Limits that result in 0/0 or ∞/∞ require algebraic simplification or L’Hôpital’s Rule (covered later).
• Continuity: A function is continuous at a point x = a if:
  • f(a) is defined
  • \lim_{x \to a} f(x) exists
  • \lim_{x \to a} f(x) = f(a)
• Common Discontinuities: Jumps, holes (removable), and vertical asymptotes (infinite).

Quick Tip

Practice Problems

1. Evaluate the limit: \lim_{x \to 2} (x^2 - 4)/(x - 2)
   Show Solution

   Step 1: Plug in x = 2 → (4 - 4)/(2 - 2) = 0/0 → indeterminate form.

   Step 2: Factor the numerator: (x^2 - 4) = (x - 2)(x + 2)

   Step 3: Cancel common factors: (x - 2)/(x - 2) → 1

   Step 4: Now evaluate the simplified expression x + 2 at x = 2

   Final Answer: 4

2. Determine whether f(x) is continuous at x = 3 where:
   f(x) = { x^2, if x ≠ 3; 9, if x = 3 }
   Show Solution

   Step 1: f(3) = 9

   Step 2: \lim_{x \to 3} x^2 = 9

   Step 3: Since limit and function value both exist and match → f is continuous at x = 3

   Final Answer: Yes, f(x) is continuous at x = 3