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## PowerPoint Slideshow about ' Mini Math Wars' - yoshio-austin

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### Mini Math Wars

### Section 1.4

Review of Limits

Windows vs. Doors

I will pick a person to answer.

If correct = +1; If incorrect other team can steal that point.

Round is over after pts awarded; question goes to other side of room.

Target Goals:

1. Be able to evaluate continuity of a function using left and right hand limits

2. Be able to apply the Intermediate Value Theorem to closed interval functions.

Continuity and One-Sided Limits

Definition of continuity

- A function f(x) is continuous at x=c if all of the following conditions exist:
- The function has a value at x=c, i.e., f(c) exists.
- The limit exists at c (check left and right hand sides).
- The limit at c = f(c).

A function is everywhere continuous if it is continuous at each point in its domain.

- A function is continuous on an interval if it is continuous at every point on that interval.

Example 1

- Is the function continuous at x=2?
- Does f(2) exist?
- Does the limit exist at x=2?
- Does the limit equal f(2)?

YES! The function is CONTINUOUS!!

YES!

3

3

YES!

YES!

Example 2

- Is the function continuous at x=2?
- Does f(2) exist?
- Does the limit exist at x=2?
- Does the limit equal f(2)?

YES!

3

5

NO!

NOT CONTINUOUS at x = 2!

Types of Discontinuities

- Jump discontinuity - the curve breaks at a particular place and starts somewhere else (non-removable)
- Point discontinuity - curve has a hole in it and a point that’s off of the curve (removable)
- Essential discontinuity - vertical asymptote (non-removable)

Ex 3) Examples of Discontinuities

IF the function is continuous, then the limit must exist!

Ex 4) Where does the following function have…

- An essential discontinuity?

At vertical asymptotes!

x = -4

- A removable discontinuity?

What can you cancel?

x = 5

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 f(c)=k.

f(a)

f (c) =

k

f(b)

c

a

b

Ex 5) Example applying IVT

Verify that the IVT applies to the given function on the given interval. Then find the value of c guaranteed by the theorem such that f(c) = 0.

1. Is f (x) continuous on [0, 3]?

2. Is 0 in the interval [f (0), f ( 3)]?

Yes

Yes

Target Goals

- Be able to evaluate continuity of a function using left and right hand limits.
- Be able to apply the Intermediate Value Theorem to closed interval functions.

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