Understanding Limits and Continuity in Calculus
Limits describe the behavior of a function as the input approaches a specific value, while continuity ensures no breaks or jumps in the function's graph at that point. To solve problems, first evaluate the limit using direct substitution: if f(x) approaches L as x approaches a, and f(a) = L, the function is continuous at a. Common techniques include factoring, rationalizing, or using L'Hôpital's rule for indeterminate forms like 0/0 or ∞/∞.
Key Principles for Evaluation
Core principles include the limit laws (sum, product, quotient) and the definition of continuity: a function f is continuous at a if lim_{x→a} f(x) = f(a). Address one-sided limits for points of discontinuity—left-hand (x→a⁻) and right-hand (x→a⁺) must match for continuity. Misconception: Infinite limits imply discontinuity; actually, vertical asymptotes indicate discontinuity but not necessarily an undefined limit.
Practical Example: Solving a Limit Problem
Consider lim_{x→2} (x² - 4)/(x - 2). Direct substitution yields 0/0, indeterminate. Factor the numerator: (x-2)(x+2)/(x-2) = x+2 for x ≠ 2. Thus, the limit is 4. For continuity, check f(2) = 4; since it matches, the function is continuous after redefining at x=2, illustrating removable discontinuity resolution.
Applications and Importance
These concepts are foundational for derivatives and integrals in calculus, used in physics for velocity (limit of displacement) and engineering for signal processing continuity. Practically, they help model real-world phenomena without abrupt changes, like population growth curves, and solving them builds problem-solving skills for advanced math and sciences.