University:
Kentucky State University
Course:
MAT 120 | Precalculus
Academic year:

2024
Views:

390

Pages:

7
Author:

Caroline Mcfarland

Continuous Function (Part 1) Nov 18, 2021 • Continuity: Definitions: f(x) is continuous (cont) at x=a if: (1) f(a) is defined (2) lim f(x) exists x→a (3) lim f(x) = f(a) x→a eg) let f(x) = 3 + e^x + x². Is f cont at x=0? (1) f(0) = 3 + e^0 + 0² = 3 + 1 + 0 = 4 (2) lim f(x) = lim (3 + e^x + x²) = 3 + 1 + 0 = 4 x→0+ x→0- f(x) cont at x=0 eg) let f(x) = { (x²-9)/(x-3), x ≠ 3 { 4, x = 3 } is f cont at x=3 or not? (1) f(3) = 4 (2) lim f(x) = lim (x²-9)/(x-3) = 0 x→3- x→3- (3) lim f(x) = lim (x²-9)/(x-3) = 6 x→3+ x→3+ f(x) ≠ lim f(x) ≠ 4 ≠ 6 => f(x) not cont at x=3 Remark: if f(x) not cont at x=a is called discontinuous at x=a (discont) eg) f(x) = |x| Cont? f(x) = { x, x≥0 {-x, x<0 } lim f(x) = 1 x→0+ lim f(x) = -1 x→0- ∴ f(x) dis cont at x=0 eg) Find the values of a & b such that f(x) = { ax + b, x ≥ 2 { x²-11, x < 2 is cont? Since f(x) cont => f(x) cont at x = 2 (1) lim f(x) = lim f(x) x→2+ x→2- (2) lim f(x) = f(2) x→2 (1) lim (ax+b) = lim (x²-11) x→2+ x→2- 4a + b = -11 (2) lim f(x) = f(2) x→2 4a + 1 = 8 + b -8 + b = -11 b = -3 4a + 1 = -11 + (-3) 4a = -15 a = -15/4 • Defn: f(x) is cont on the open interval (a,b) if f(x) is cont for all c ∈ (a,b) • f(x) is cont on (a,b) or [a,b] or (a,∞) or (-∞,b) or (-∞,∞) • Polynomials are continuous everywhere. Eg) let f(x) = |x|, show that f(x) is cont everywhere. |x| = { x, x ≥ 0 {-x, x < 0 } lim f(x) = f(0) → cont at f(0) x→0+ lim f(x) = f(0) → cont at f(0) x→0- ∴ f(x) cont everywhere. • Remark: If $f(x) = \frac{p(x)}{q(x)}$ where p(x), q(x) are polynomials, then f(x) is cont everywhere except when q(x) = 0 eg) $f(x) = \frac{x^3 + 3}{x^3 - x}$ cont everywhere - x³ - x = 0 x(x² - 1) = 0 x(x-1)(x+1) = 0 eg) if $f(x) = \frac{x+3}{x²+ax+1}$ find the values of a that makes f(x) cont everywhere. x²+ax+1 ≠ 0 b²-4ac < 0 a²-4(1)(1) < 0 a² - 4 < 0 (a+2)(a-2) < 0 ∴ a ∈ (-2, 2) • Defn: 1. f(x) is cont at x = c from the right if lim f(x) = f(c) 2. f(x) is cont at x = c from the left if lim f(x) = f(c) 3. f(x) is cont on (a,b) if: a. f(x) is cont on (a,b) b. f(x) is cont at a from the right c. f(x) is cont at b from the left