Lesson Plan Outline\

Activities:

1. Grouping Activity – students will get into groups of 3 - 4

2. Discovering Limits – group work

3. Notes – intro to limits by graphing

4. Think-Pair-Share – solving limits by graphing worksheet

Warm Up: Trig Quiz – unit circle, identities, solving

Reflection:

Objective: Students will be able to define and find limits by looking at the graph of a function.

Teacher: Burton Date:Class:Calc Period:

Closure: Assign HW

DOL:

Attitudes & Perceptions

Acquire & Integrate

Knowledge

Extend & Refine Knowledge

Use knowledge meaningfully

Productive Habits of Mind

READING STRATEGIES:

Word Wall

Notes – Intro to limits

Importance of Limits:1. Limits allow us to work with _______________________that we didn’t have the

________________ to consider previously

EX: functions with ______________________________

functions that ______________________________

these are the most important functions for modern applications of mathematics.

2. Limits allow us to work more effectively with the _____________________________________

____________________________________________________.

3. The _________________________________________ is built on limits. All calculus

_______________________ and the definitions of derivatives and integrals use limits (if just

implicitly).

Motivating Example 1:f(x) =

What are the values for y when x=0, 1, -1, -2?

What is interesting about the graph?

Where are the discontinuities?

Intuitive definition of Limit (version 1):

Left- and right-sided limit notation and limits involving infinity.

Revisiting Example 1:

Evaluate

a. g.

b. h.

c. i.

d. j.

e. k.

f. l.

Important Note involving right- and left-sided limits:

if and only if ________________ and _______________

Intuitive Definition of Limit (version 2, from Stewart):

Reasons that Limits Fail to Exist at a Point:

Summary and extension:

Whether or not a the limit of a function exists at a point, , has __________ to do with whether or not

___________ exists. In other words, limits can _______________ holes in the function.

Practice: p. 79 #1 – 15 odd

Notes – Intro to limits

Importance of Limits:1. Limits allow us to work with FUNCTIONS that we didn’t have the

TOOLS to consider previously

EX: functions with DISCONTINUITIES

functions that “ BLOW UP” (GO TO INFINITY)

these are the most important functions for modern applications of mathematics.

2. Limits allow us to work more effectively with the IMPOSSIBLY LARGE AND SMALL VALUES THAT OCCUR IN THESE FUNCTIONS.

3. The FOUNDATION OF CALCULUS is built on limits. All calculus

THEOREMS and the definitions of derivatives and integrals use limits (if just

implicitly).

Motivating Example 1:f(x) =

What are the values for y when x=0, 1, -1, -2?

What is interesting about the graph?

Where are the discontinuities?

Intuitive definition of Limit (version 1):

The limit of a function is the value that the function gets arbitrarily close to as x gets arbitrarily close to some number. In other words, when x gets close to some number (x0), then y gets close to some number (L). We call L the limit.

Left- and right-sided limit notation and limits involving infinity.

means we look at f(x) from the RIGHT of x = 1.

means we look at f(x) from the LEFT of x = 1.

Revisiting Example 1:

Evaluate

a. g.

b. h.

c. i.

d. j.

e. k.

f. l.

Important Note involving right- and left-sided limits:

if and only if and

Reasons that Limits Fail to Exist at a Point (keep in mind the example):

Summary and extension:

Whether or not a the limit of a function exists at a point, , has NOTHING to do with whether or not

f(c) exists. In other words, limits can IGNORE holes in the function.

Homework: Stewart, pages 79–81, numbers 4–10, 13

Go over left- and right-sided limit notation and limits involving infinity.

Important Note: if and only if and

Group Practice: from Anton, exercise set 2.1. Students should write the important note at the top.

Intuitive Definition of Limit (version 2, from Stewart):

We write and say “the limit of as approaches , equals ” if we can make

the values of arbitrarily close to (as close to as we like) by taking to be sufficiently close

to (on either side of ).

Brainstorm Reasons that Limits Fail to Exist at a Point:The one-sided limits aren’t equal to the same limit at that point.The function becomes infinite at that point (more on this later).The function oscillates wildly at that point (more on this later).

Summary:Whether or not a the limit of a function exists at a point, , has nothing to do with whether or not exists. In other words, limits can ignore holes in the function.

Homework: Stewart, pages 79–81, numbers 4–10, 13