2.2 the concept of limit wed sept 16 do now sketch the following functions. describe them
TRANSCRIPT
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