Approaching a Value: Limits & Continuity
Limits describe what value a function approaches as the input gets closer and closer to a particular number, even if the function never actually reaches that value. Continuity means a function has no breaks, jumps, or holes at a point. Together, limits and continuity form the theoretical foundation for derivatives and integrals — the two pillars of calculus.
Components of Limits & Continuity
This section covers the key concepts:
- Evaluating Limits: Finding the value a function approaches as x → a, using direct substitution, factoring, or rationalizing.
- One-Sided Limits: The limit from the left (x → a⁻) and the limit from the right (x → a⁺) may differ; the two-sided limit exists only when both are equal.
- Limits at Infinity: Describing function behavior as x grows without bound, revealing horizontal asymptotes.
- Continuity: A function is continuous at x = a if f(a) is defined, the limit as x → a exists, and the limit equals f(a).
Examples of Limits & Continuity
Evaluating Limits Examples
- Find lim(x→3) of (2x + 1): Direct substitution gives 2(3) + 1 = 7.
- Find lim(x→2) of (x² - 4)/(x - 2): Factor the numerator as (x-2)(x+2), cancel (x-2), leaving lim(x→2) of (x+2) = 4.
- Find lim(x→0) of sin(x)/x: This is a well-known limit that equals 1 (used frequently in calculus proofs).
One-Sided Limits Examples
- For f(x) = |x|/x, the left limit as x → 0⁻ is -1 and the right limit as x → 0⁺ is 1. Since they differ, the two-sided limit does not exist.
- For a step function that equals 2 when x < 1 and 5 when x ≥ 1, the left limit at x = 1 is 2 and the right limit is 5.
- For f(x) = √x, the left limit as x → 0⁻ does not exist (square root of negatives is undefined), but the right limit as x → 0⁺ is 0.
Limits at Infinity Examples
- Find lim(x→∞) of 3/x: As x grows, 3/x shrinks to 0. The horizontal asymptote is y = 0.
- Find lim(x→∞) of (2x + 1)/(x - 3): Divide top and bottom by x to get (2 + 1/x)/(1 - 3/x), which approaches 2/1 = 2.
- Find lim(x→∞) of (5x²)/(x² + 4): Divide by x² to get 5/(1 + 4/x²), which approaches 5.
Continuity Examples
- f(x) = x² is continuous everywhere — you can draw the parabola without lifting your pencil.
- f(x) = 1/x is not continuous at x = 0 because f(0) is undefined (division by zero).
- The function defined as f(x) = x + 1 when x ≠ 2 and f(2) = 5 has a discontinuity at x = 2 because the limit (which is 3) does not equal f(2) = 5.