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Section 1.4 – Continuity and One-Sided Limits

Section 1.4 – Continuity and One-Sided Limits. Example. When x =5, all three pieces must have a limit of 8. . Find values of a and b that makes f ( x ) continuous. . Continuity at a Point. f ( x ). L. x. c. For every question of this type, you need (1), (2), (3), conclusion.

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Section 1.4 – Continuity and One-Sided Limits

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  1. Section 1.4 – Continuity and One-Sided Limits

  2. Example When x=5, all three pieces must have a limit of 8. Find values of a and b that makes f(x) continuous.

  3. Continuity at a Point f(x) L x c For every question of this type, you need (1), (2), (3), conclusion. A function f is continuous at c if the following three conditions are met: • is defined. • exists.

  4. Example 1 The function is clearly defined at x= 0 With direct substitution the limit clearly exists at x=0 The value of the function clearly equals the limit at x=0 f is continuous at x = 0 Show is continuous at x = 0.

  5. Example 2 The function is clearly 10 at x = 2 With direct substitution the limit clearly exists at x=0 The behavior as x approaches 2 is dictated by 8x-1 The value of the function clearly does not equal the limit at x=2 f is not continuous at x = 2 Show is not continuous at x = 2.

  6. Discontinuity Typically a hole in the curve Step/Gap Asymptote If f is not continuous at a, we say f is discontinuous at a, or f has a discontinuity at a.

  7. Example Find the x-value(s) at which is not continuous. Which of the discontinuities are removable? If f can be reduced, then the discontinuity is removable: Notice that: This is the same function as f except at x=-3 There is a discontinuity at x=-3 because this makes the denominator zero. f has a removable discontinuity at x = -3

  8. Indeterminate Form: 0/0 Let: f(x) L If: x c Then f(x) has a removable discontinuity at x=c. If f(x) has a removable discontinuity at x=c. Then the limit of f(x) atx=c exists.

  9. One-Sided Limits: Left-Hand If f(x) becomes arbitrarily close to a single REAL number L as x approaches c from values less than c, the left-hand limit is L. f(x) L x c The limit of f(x)… is L. Notation: as x approaches c from the left…

  10. One-Sided Limits: Right-Hand If f(x) becomes arbitrarily close to a single REAL number L as x approaches c from values greater than c, the right-hand limit is L. f(x) L x c The limit of f(x)… is L. Notation: as x approaches c from the right…

  11. Example 1 Evaluate the following limits for

  12. Example 2 Analytically find .

  13. The Existence of a Limit f(x) L x c A limit exists if… Left-Hand Limit = Right-Hand Limit Let fbe a function and let c be real numbers. The limit of f(x)as x approaches c is Lif and only if

  14. Example 1 Use when x>2 Use when x<2 You must use the piecewise equation: Analytically show that .

  15. Example2 You must use the piecewise equation: Use when x>-1 Use when x<-1 Analytically show that is continuous at x = -1.

  16. Continuity on a Closed Interval f(b) f(a) x a b Must have closed dots on the endpoints. A function f is continuous on [a, b]if it is continuous on (a, b) and

  17. Example 1 Use the graph of t(x) to determine the intervals on which the function is continuous.

  18. Example 2 Are the one-sided limits of the endpoints equal to the functional value? By direct substitution: The domain of f is [-1,1]. From our limit properties, we can say it is continuous on (-1,1) f is continuous on [-1,1] Discuss the continuity of Is the middle is continuous?

  19. Properties of Continuity • Scalar Multiple: • Sum/Difference: • Product: • Quotient: if • Composition: • Example: Since are continuous, is continuous too. If b is a real number and f and g are continuous at x = c, then following functions are also continuous at c:

  20. Intermediate Value Theorem This theorem does NOT find the value of c. It just proves it exists. f(b) k f(a) c a b If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that:

  21. Free Response Exam 2007 Notice how every part of the theorem is discussed (values of the function AND continuity). We will learn later that this implies continuity. Since h(3) < -5 < h(1) and h is continuous, by the IVT, there exists a value r, 1 < r < 3, such that h(r) = -5.

  22. Example Find an output less than zero Find an output greater than zero Since f(0) < 0 and f(2) > 0 There must be some csuch that f(c) = 0 by the IVT The IVT can be used sincef is continuous on [-∞,∞]. Use the intermediate value theorem to show has at least one root.

  23. Example Solve the equation for zero. Find an output greater than zero Find an output less than zero Since and The IVT can be used sincethe left and right side are both continuous on [-∞,∞]. There must be some csuch that cos(c) = c3 - c by the IVT Show that has at least one solution on the interval .

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