Section 1.4 – Continuity and One-Sided Limits

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

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

Example 1 Show is continuous at x = 0. f is continuous at x = 0 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

Example 2 Show is not continuous at x = 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

Discontinuity If f is not continuous at a, we say f is discontinuous at a, or f has a discontinuity at a. Types Of Discontinuities Removable Able to remove the “hole” by defining f at one point Non-Removable NOT able to remove the “hole” by defining f at one point Typically a hole in the curve Step/Gap Asymptote

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

Indeterminate Form: 0/0 Let: If: 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) at x=c exists.

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. c L f(x) x The limit of f(x)… is L. Notation: as x approaches c from the left…

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. c L f(x) x The limit of f(x)… is L. Notation: as x approaches c from the right…

Example 1 Evaluate the following limits for

Example 2 Analytically find .

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

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

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

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

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

Is the middle is continuous? Example 2 Discuss the continuity of 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] Is the middle is continuous?

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

Intermediate Value Theorem 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: a b f(a) f(b) k This theorem does NOT find the value of c. It just proves it exists. c

We will learn later that this implies continuity. 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.

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

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