Post on 30-Dec-2015
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