Chapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the

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1 Chapter 2 Limits and Continuity 2.1 Rates of change and Tangents to Curves Definition : interval [x 1, x 2 ] is The average Rate of change of y = f(x) with respect to x over the y x = f(x 2) f(x 1 ) = f(x 1 + h) f(x 1 ), h 0 x 2 x 1 h = slope of secant to the graph of f Definition : Slope of a curve y = f(x) at p(x 1, y 1 ) is the slope of the tangent line at p, f(x 1 + h) f(x 1 ) slope = lim. h 0 h 11

2 2.2 Limits of functions and Limits laws Definition : Limits of a function : Let f(x) defined on an open interval about x 0, except possibly at x 0 itself, if f(x) is arbitrarily close to L for all x sufficiently close to x 0, we say that f approaches the limit L as x approaches x 0. and write lim x x0 f(x) = L Theorem : The Limit Laws: Let L, M, c, k be real numbers and lim x c f(x) = L, lim x c g(x) = M then: 1) lim x c [f(x) ± g(x)] = L ± M. 2) lim x c [kf(x) = kl. 3) lim x c [f(x)g(x) = LM. 4) lim x c f(x) g(x) = L M, M 0. 5) lim x c [f(x)] n = L n, n is positive integer. 6) lim x c n f(x) = n L, nɛn. ( If n is even, we assume that lim x c f(x) = L > 0. 12

3 Theorem : Limits of Polynomials: If P (x) = a n x n + a n 1 x n a 0 Then lim x c P (x) = P (c) = a n c n + a n 1 c n a 0. Theorem : Limits of Rational functions : If P (x) and Q(x) are polynomials and Q(c) 0 then: P (x) lim x c = P (c). Q(x) Q(c) Eliminating zero Denominators Algebraically: Finding limits by multiplying both numerator and denominator by the conjugate : Theorem : The Sandwich theorem: Suppose that g(x) f(x) h(x), x in some open interval containing c, except possibly at x = c itself. Suppose also that lim x c g(x) = lim x c h(x) = L, Then: lim x c f(x) = L 13

4 Theorem Suppose that f(x) g(x), x in some open interval containing c, except possibly at x = c itself and the limits of f, g both exist as x c Then: lim x c f(x) lim x c g(x). 14

5 2.3 The precise definition of a limit Definition : Let f(x) be defined on an open interval about x 0, except possibly at x 0 itself, we say that lim x x0 f(x) = L if ɛ > 0, δ > 0 such that x, 0 < x x 0 < δ f(x) L < ɛ 15

6 2.4 One sided Limits Definition : 1) Let f(x) be defined on (c, b), let x approaches c from within that interval, then f approaches L and we say that f has a right -hand limit L at c. and write: lim x c + f(x) = L. 2) If f(x) defined on (a, c) and x approaches c from within that interval, f(x) approaches M then f has a left -hand limit M at c and write lim x c f(x) = M Find limits Theorem : A function f(x) has a limit as x c it has left -hand and righthand limits their and those one sided limits are equal. lim x c f(x) = L lim x c + f(x) = L and lim x c f(x) = L Precise Definition of one sided Limits Definition ) lim x c + f(x) = L if, ɛ > 0, δ > 0 such that x, c < x < c + δ f(x) L < ɛ 2) lim x c f(x) = L if, ɛ > 0, δ > 0 such that x, c δ < x < c f(x) L < ɛ 16

7 Theorem : lim θ 0 sin θ θ = 1, θ in radian. 17

8 2.5 Continuity Definition : Continuity at an Interior Point: A function y = f(x) is continuous at an interior point c of its domain if lim x c f(x) = f(c) Continuity at End Point: A function y = f(x) is continuous at a left end point a or continuous at a right end point b of its domain if lim x a + f(x) = f(a) or lim x b f(x) = f(b) respectively. If f is not continuous at a point c, we say f is discontinuous at c, and c is a point of discontinuity of f. Continuity Test: A function f(x) is continuous at an interior point x = c of its domain it meets the following three conditions : (1) f(c) exist. (2) lim x c f(x) exist. (3) lim x c f(x) = f(c). Types of discontinuity : 1) Removable Discontinuity : The function has a limit at a point c and we can remove the discontinuity by redefine the function lim x c f(x) = f(c). 2) Jump discontinuity : The one sided limits exist but have different values. 3) Infinite Discontinuity : the function has an infinite limit as x c 4) Oscillating Discontinuity : the function oscillate as x c 18

9 Continuity on an interval : The function is continuous on an interval it is continuous at every point of the interval. Note : A continuous function is the function that is continuous at every point of its domain. Theorem Properties of continuous functions: If the functions f, g are continuous at x = c then the following are continuous at c. f f ± g, kf, fg,, g(c) 0, f n, nɛn g and n f provided it is defined on an open interval containing c, nɛn. 19

10 Theorem : If f is continuous at c and g is continuous on f(c) then g f is continuous at c. Theorem :Limits of continuous functions: If g is continuous at b and lim x c f(x) = b then lim x c g(f(x)) = g(lim x c f(x)) = g(b). Theorem Intermediate value theorem (IV T ) : If f is continuous function on a closed interval [a, b], and if y 0 f(a), f(b) then y 0 = f(c) for some c in [a, b]. is any value between Note that : 1) If f is not continuous at a point between a, b then we can find y 0 f(c) 2) IV T tells us that if f is continuous then any interval on which f changes sign, contains a zero of the function. That is, y 0 = 0 between f(a), f(b) cɛ[a, b] such that f(c) = 0 We call this a solution or root of the equation f(x) = 0. 20

11 2.6 Limits involving infinity ; Asymptotes of graphs Definition ) We say that f(x) has the limit L as x and write lim x f(x) = L if ɛ > 0, a corresponding number M such that x > M, f(x) L < ɛ. 2) We say that f(x) has the limit L as x and write lim x f(x) = L if ɛ > 0, a corresponding number N such that x < N, f(x) L < ɛ. Show that : Theorem : All the limit laws are true when we replace x c by x ± Limits at infinity of rational functions : To determine the limit of rational function as x ±, we first divide the numerator and denominator by the highest power of x in the denominator 21

12 Horizontal Asymptotes :(H.A) A line y = b is a H.A of the graph of a function y = f(x) if either lim x f(x) = b or lim x f(x) = b If both limits hold we say that f has a H.A from both sides right and left. 22

13 Oblique Asymptotes :(O.A) If the degree of numerator of rational function is 1 greater than the degree of the denominator, the graph has an oblique asymptote. We find an equation for the asymptote by dividing numerator by denominator to express f as linear function plus a remainder that goes to zero as x ± Infinite limits : lim x x 2 =, lim x 0 1 x 2 = in each case the limit does not exist. Precise Definition of limits : 1) lim x x0 f(x) = if B > 0, δ > 0 such that x, 0 < x x 0 < δ f(x) > B. 2) lim x x0 f(x) = if B < 0, δ > 0 such that x, 0 < x x 0 < δ f(x) < B. 1 Prove that lim x 0 = x 2 Vertical Asymptotes :(V.A) A Line x = a is vertical asymptote of the graph of f if either lim x a + f(x) = ± or lim x a f(x) = ± Find H.A and V.A of the following functions: 23

14 Dominant Terms : To know how f behaves near a point or when x large f(x) = x2 +1 x 1 = (x + 1) + 2 x 1 We see that f behaves like (x + 1) as x and it behaves like 2 x 1 as x 1. 1) Graph the rational function Include the graphes equations of asymptotes and Dominant terms. y = 2x x+1 2) Graph giving the following conditions: f(1) = 1, f( 1) = 0, lim x f(x) = 0, lim x 0 + f(x) =, lim x 0 f(x) =, lim x f(x) = 1. 24

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