Limit s MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007
Bounded Functions Definition Let A R, let f : A R, and let c R be a cluster point of A. We say that f is bounded on a neighborhood of c if there exists a δ-neighborhood V δ (c) of c and a constant M > 0 such that we have f(x) M for all x A V δ (c). If A R and f : A R has a limit at c R, then f is bounded on some neighborhood of c.
Bounded Functions Definition Let A R, let f : A R, and let c R be a cluster point of A. We say that f is bounded on a neighborhood of c if there exists a δ-neighborhood V δ (c) of c and a constant M > 0 such that we have f(x) M for all x A V δ (c). If A R and f : A R has a limit at c R, then f is bounded on some neighborhood of c.
Bounded Functions Definition Let A R, let f : A R, and let c R be a cluster point of A. We say that f is bounded on a neighborhood of c if there exists a δ-neighborhood V δ (c) of c and a constant M > 0 such that we have f(x) M for all x A V δ (c). If A R and f : A R has a limit at c R, then f is bounded on some neighborhood of c.
Algebra of Functions Definition Let A R and let f and g be functions defined on A to R. We define the sum f + g, the difference f g, and the product fg on A to R to be the functions given by (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (fg)(x) = f(x)g(x), for all x A. Further, if b R, we define the multiple bf to be the function given by (bf)(x) = bf(x) for all x A. Finally, if h(x) 0 for x A, we define the quotient f/h to be the function given by ( ) f (x) = f(x) for all x A. h h(x)
Algebra of Limits Let A R, let f and g be functions on A to R and let c R be a cluster point of A. Further, let b R. 1 If lim f(x) = L and lim g(x) = M, then lim (f + g)(x) = L + M lim (fg)(x) = LM lim lim (f g)(x) = L M (bf)(x) = bl. 2 If h : A R and h(x) 0 for all x A, and if lim h(x) = H 0, then ( ) f lim (x) = L h H.
Algebra of Limits Let A R, let f and g be functions on A to R and let c R be a cluster point of A. Further, let b R. 1 If lim f(x) = L and lim g(x) = M, then lim (f + g)(x) = L + M lim (fg)(x) = LM lim lim (f g)(x) = L M (bf)(x) = bl. 2 If h : A R and h(x) 0 for all x A, and if lim h(x) = H 0, then ( ) f lim (x) = L h H.
Examples Example 1 lim x 2 (x 2 + 1)(x 3 4) x 3 4 2 lim x 2 x 2 + 1 x 4 4 3 lim x 2 3x 6 4 If p is a polynomial function, then lim p(x) = p(c). 5 If p and q are polynomial functions on R and if q(c) 0, then p(x) lim q(x) = p(c) q(c).
Squeeze Let A R, let f : A R and let c R be a cluster point of A. If a f(x) b for all x A with x c, and if lim f(x) exists, then a lim f(x) b.
Squeeze Let A R, let f : A R and let c R be a cluster point of A. If a f(x) b for all x A with x c, and if lim f(x) exists, then a lim f(x) b.
Squeeze (cont.) (Squeeze ) Let A R, let f, g, h : A R and let c R be a cluster point of A. If f(x) g(x) h(x) for all x A with x c, and if lim f(x) = L = lim h(x), then lim g(x) = L.
Squeeze (cont.) (Squeeze ) Let A R, let f, g, h : A R and let c R be a cluster point of A. If f(x) g(x) h(x) for all x A with x c, and if lim f(x) = L = lim h(x), then lim g(x) = L.
Examples Example 1 lim x 0 sin x = 0 2 lim cos x = 1 x 0 ( ) 3 cos x 1 lim = 0 x 0 x ( ) 4 sin x lim = 1 x 0 x ( ) 5 1 lim x sin = 0 x 0 x
Final Result Let A R, let f : A R and let c R be a cluster point of A. If [ ] lim f(x) > 0 respectively, lim f(x) < 0, then there exists a neighborhood V δ (c) of c such that f(x) > 0 [respectively, f(x) < 0] for all x A V δ (c), x c.
Final Result Let A R, let f : A R and let c R be a cluster point of A. If [ ] lim f(x) > 0 respectively, lim f(x) < 0, then there exists a neighborhood V δ (c) of c such that f(x) > 0 [respectively, f(x) < 0] for all x A V δ (c), x c.
Homework Read Section 4.2 Pages 110-111: 1, 3, 4, 5, 9, 11, 12 Boxed problems should be written up separately and submitted for grading at class time on Friday.