2.2 The Concept of LimitWed Sept 16

Do Now

Sketch the following functions. Describe them.

HW: p.64 #1 5 7

• 1) a- 11.025 m

b- 22.05 m/s

c- ~19.6 m/s

• 5) 0.3 m/s

• 7) a- dollars/year

b- [0, 0.5]: 7.8461; [0, 1]: 8

c- ~ $8/year

Concept of a Limit

• Take note of the two functions.

• Both functions are undefined at x = 2

• Let’s take a look at each function for x close to 2

F(x) and g(x)

X

F(x) G(x)

1.9 13.9 3.9

1.99 103.99 3.99

1.999 1003.999 3.999

1.9999 10003.9999 3.9999

1.99999 100003.9999 3.99999

1.999999 1000004 3.999999

Limits

• We consider the limit of a function as the value the function approaches as it gets closer to a certain value.

• In the table, we approached x = 2 from the left side. We denote this as

Limits Cont’d

• Repeat the same process from the right side

One-sided Limits

• Limits from the left or right side of a function are called one-sided limits

• If two one-sided limits of f(x) are the same, they comprise the limit of f(x)

• Important: A limit exists if and only if both one-sided limits exist and are equal.

Limits and Graphs

• For now, we’ll be using graphs and tables to see if limits exist or not

• A graphing calculator helps when looking at functions to determine where the limits exist

Exs in book

Canceling Factors

• If a function has identical factors in the numerator and denominator, they can be cancelled before finding the limit

• Canceling factors will not affect the limit of the function

Piecewise functions

• When looking at piecewise functions, it is often important to use one-sided limits to determine if a limit exists.

• Absolute value functions are included in this idea

Closure

• What is a one-sided limit?

• What is the notation involved with limits?

• How do we know if a limit exists? What must be true?

• Homework: pp 74-75 #1, 3, 5, 6, 38, 47, 53

2.2 Limits using Graphs/TablesThurs Sep 17

• Do Now

• Let f(x) = x + 3 g(x) = 4 / (x - 3)

• 1) Find

• 2) Find

HW Review: p.74 #1 3 5 6 38 47 53

• 1) 3/2 47) c = 2 (inf, inf)• 3) 3/5 c= 4 (- inf, 10)• 5) 1.5 v.a. x = 2• 6) 1.5 53) c = 1 (3, 3)• 38) c = 1 (3, 1) DNE c = 3 (-inf, 4)

c = 2 (2, 1) DNE c = 5 (2, -3)

c = 4 (2, 2) exists c = 6 (inf, inf)

Review

• A limit is the y-value a function approaches as x gets close to something

• It does NOT matter what the function is AT that point…only what it seems to approach!

How to compute limits?

• For now, we can use either a graph or a table to determine a function’s limit

• Use tables when it is difficult to determine where a graph is approaching (not whole numbers)

Practice

• Worksheet

• 2.2 Quiz tomorrow– Limits

• Graphs• Table

Closure

• Graph and find

• HW: Finish selected problems on worksheet

• Quiz tomorrow

2.2 QuizFri Sept 18

• Do Now

• Find the left and right hand limits of

HW Review: worksheet p.110-111 #1-12

2.2 Quiz