Limit Theorems. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Limit Theorems

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1 Limit s MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007

2 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.

3 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.

4 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.

5 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)

6 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.

7 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.

8 Examples Example 1 lim x 2 (x 2 + 1)(x 3 4) x lim x 2 x x 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).

9 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.

10 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.

11 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.

12 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.

13 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

14 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.

15 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.

16 Homework Read Section 4.2 Pages : 1, 3, 4, 5, 9, 11, 12 Boxed problems should be written up separately and submitted for grading at class time on Friday.

Evaluating Limits Analytically. By Tuesday J. Johnson

Evaluating Limits Analytically By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews