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Limit and Continuity

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2.1 Rate of Change and Limits (1) Average and Instantaneous Speed y

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2.1 Rate of Change and Limits (2, Example 2) Average and Instantaneous Speed y t=2 v=? When different value of h

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2.1 Rate of Change and Limits (3) Definition of Limit c L x y

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2.1 Rate of Change and Limits (4) Definition of Limit

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2.1 Rate of Change and Limits (5) Properties of Limit This can be applied to do the limits of all polynominal and rational functions.

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2.1 Rate of Change and Limits (6, Theorem 1) Properties of Limit

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2.1 Rate of Change and Limits (7, Theorem 1) Properties of Limit

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2.1 Rate of Change and Limits (8, Theorem 1) Properties of Limit

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2.1 Rate of Change and Limits (9, Example 3) Properties of Limit

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2.1 Rate of Change and Limits (10,Theorem 2) Properties of Limit

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2.1 Rate of Change and Limits (11, Example 4) Properties of Limit

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2.1 Rate of Change and Limits (12, Example 5) Properties of Limit 2-1 Exercise 63

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2.1 Rate of Change and Limits (13, Example 6) Properties of Limit

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2.1 Rate of Change and Limits (14) One-sided and Two-sided Limits c x f(x) f(c + ) f(c - )

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2.1 Rate of Change and Limits (15, Example 7) One-sided and Two-sided Limits

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2.1 Rate of Change and Limits (16, Theorem 3) One-sided and Two-sided Limits

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2.1 Rate of Change and Limits (17, Example 8) One-sided and Two-sided Limits

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2.1 Rate of Change and Limits (18, Theorem 4) Sandwich Theorem x y c L f g h

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2.1 Rate of Change and Limits (19, Example 9) Sandwich Theorem h(x) = x 2 g(x) = -x 2 f(x) = x 2 sin(1/x)

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2.2 Limits Involving Infinite (1) Finite Limits as x→± The symbol of infinite ( ) does not represent a real number. The use to describe the behavior of a function when the values in its domain or range out grow all finite bounds.

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2.2 Limits Involving Infinite (2) Finite Limits as x→±

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2.2 Limits Involving Infinite (3) Finite Limits as x→±

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2.2 Limits Involving Infinite (4, Example 1) Finite Limits as x→±

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2.2 Limits Involving Infinite (5, Example 2) Sandwich Theorem Revisited

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2.2 Limits Involving Infinite (6, Theorem 5-1) Sandwich Theorem Revisited

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2.2 Limits Involving Infinite (7, Theorem 5-2) Sandwich Theorem Revisited

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2.2 Limits Involving Infinite (8, Example 3) Sandwich Theorem Revisited

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2.2 Limits Involving Infinite (9, Exploration 1-1) Sandwich Theorem Revisited

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2.2 Limits Involving Infinite (10, Exploration 1-2) Sandwich Theorem Revisited

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2.2 Limits Involving Infinite (11, Exploration 1-3) Sandwich Theorem Revisited

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2.2 Limits Involving Infinite (12) Infinite Limits as x →a

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2.2 Limits Involving Infinite (13) Infinite Limits as x →a

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2.2 Limits Involving Infinite (14, Example 4) Infinite Limits as x →a

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2.2 Limits Involving Infinite (15, Example 5) Infinite Limits as x →a

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2.2 Limits Involving Infinite (16, Example 6) End Behavior Models

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2.2 Limits Involving Infinite (17) End Behavior Models

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2.2 Limits Involving Infinite (18, Example 7) End Behavior Models

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2.2 Limits Involving Infinite (19) End Behavior Models IF one function provides both a left and right behavior model, it called an end behavior model. In general, g(x) = a n x n ia an end behavior model for the polynominal function f(x) = a n x n + a n-1 x n-1 +…+ a o. In the large, all polynominals behave like monomials. This is the key to the end behavior of rational functions.

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2.2 Limits Involving Infinite (20, Example 8) End Behavior Models

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2.2 Limits Involving Infinite (21, Example 9) End Behavior Models

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2.2 Limits Involving Infinite (22, Example 10) Seeing Limits as x→±

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2.3 Continuity (1, Example 1-1) Continuity at a Point

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2.3 Continuity (2, Example 1-2) Continuity at a Point

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2.3 Continuity (3) Continuity at a Point c Continuity from both side Continuity from the left b a Continuity from the right

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2.3 Continuity (4, Example 2) Continuity at a Point

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2.3 Continuity (5) Continuity at a Point y = f(x) 1 continuous at x=0 y = f(x) 1 continuous at x=0 If it had f(0)=1 y = f(x) 1 2 continuous at x=0 If it had f(0)=1 continuity at x = 0 are removable

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2.3 Continuity (6) Continuity at a Point y = f(x) 1

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2.3 Continuity (7) Continuity at a Point

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2.3 Continuity (8, Exploration1-1,2) Continuity at a Point

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2.3 Continuity (9, Exploration1-3) Continuity at a Point

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2.3 Continuity (10, Exploration1-4) Continuity at a Point

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2.3 Continuity (11, Exploration1-5) Continuity at a Point

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2.3 Continuity (12) Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain.

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2.3 Continuity (12, Example 3) Continuous Functions

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2.3 Continuity (13) Continuous Functions Polynominal functions f are continuous at every real number c because lim x → c, f(x) = f(c). Rational functions are continuous at every point of their domains. They have points of discontinuity at the zeros of their denominators. The absolute value function y = |x| is continuous at every real number. The exponential, logarithmic, trigonometric functions and radical function like y = x (1/n) (n = a positive integer greater than 1) are continuous at every point of their domains.

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2.3 Continuity (14, Theorem 6) Algebraic Combinations

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2.3 Continuity (15, Theorem 7) Composites c f(c)f(c) continuous at c g(f(c)) continuous at f(c)

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2.3 Continuity (16, Example 4) Composites

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2.3 Continuity (17, Theorem 8) Intermediate Value Theorem for Continuous Functions a b f(a)f(a) f(b)f(b) c yoyo

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2.3 Continuity (18, Example 5) Intermediate Value Theorem for Continuous Functions

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2.4 Rates of Change and Tangent Lines (1) Average Rates of Change The average rate of change of a quantity over a period of time is the amount of change divided by the time it takes. Such as average speed, grows rate of populations, monthly rainfall. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval. f(x) x y dx dy

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2.4 Rates of Change and Tangent Lines (2) Average Rates of Change (Example 1)

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2.4 Rates of Change and Tangent Lines (3) Average Rates of Change (Example 2-1)

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2.4 Rates of Change and Tangent Lines (4) Average Rates of Change (Example 2-2)

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2.4 Rates of Change and Tangent Lines (5) Average Rates of Change (Example 2-3)

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2.4 Rates of Change and Tangent Lines (6) Average Rates of Change (Example 2-4)

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2.4 Rates of Change and Tangent Lines (7) Tangent to Curve The rate at which the value of the function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. Are we to define the tangent line at an arbitrary point P on the curve and find its slope from the function y = f(x) ? Our usual definition of slope requires two points

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2.4 Rates of Change and Tangent Lines (8) Tangent to Curve The procedure to find the slope of a point P is follows : 1.We start with what we can calculate, namely, the slope of a secant through P and a point Q nearby on the curve. 2.We find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. 3.We define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.

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2.4 Rates of Change and Tangent Lines (9) Tangent to Curve (Example 3-1) P(2,4) Q(2+h, (2+h) 2 )

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2.4 Rates of Change and Tangent Lines (10) Tangent to Curve (Example 3-2) P(2,4) Q(2+h, (2+h) 2 )

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2.4 Rates of Change and Tangent Lines (11) Tangent to Curve (Example 3-3) P(2,4) Q(2+h, (2+h) 2 )

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2.4 Rates of Change and Tangent Lines (12) Slope of a Curve a a+h x y h y=f(x) f(a+h)-f(a) P(a, f(a)) Q(a+h, f(a+h))

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2.4 Rates of Change and Tangent Lines (13) Tangent to Curve (Example 4-a)

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2.4 Rates of Change and Tangent Lines (14) Tangent to Curve (Example 4-b)

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2.4 Rates of Change and Tangent Lines (15) Tangent to Curve (Example 4-c)

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2.4 Rates of Change and Tangent Lines (16) Tangent to Curve

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2.4 Rates of Change and Tangent Lines (17) Normal to a Curve (Example 5)

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2.4 Rates of Change and Tangent Lines (18) Speed Revisited (Example 6) A body’s average speed along a coordinate axis for a given period of time is the average rate of change of its position y = f(t). Its instantaneous speed at ant time t is the instantaneous rate of change of position with respect to time at time t.

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