Section 1 4 continuity and one sided limits
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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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Section 1 4 continuity and one sided limits

Section 1.4 – Continuity and One-Sided Limits


Example

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

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.


Example 1

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.


Example 2

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.


Discontinuity

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.


Example1

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

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.


One sided limits left hand

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…


One sided limits right hand

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…


Example 11

Example 1

Evaluate the following limits for


Example 21

Example 2

Analytically find .


The existence of a limit

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


Example 12

Example 1

Use when x>2

Use when x<2

You must use the piecewise equation:

Analytically show that .


Example2

Example2

You must use the piecewise equation:

Use when x>-1

Use when x<-1

Analytically show that is continuous at x = -1.


Continuity on a closed interval

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


Example 13

Example 1

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


Example 22

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?


Properties of continuity

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:


Intermediate value theorem

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:


Free response exam 2007

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.


Example3

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.


Example4

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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