a2 determining limits using limit laws and algebra n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
A2 – Determining Limits using Limit Laws and Algebra PowerPoint Presentation
Download Presentation
A2 – Determining Limits using Limit Laws and Algebra

Loading in 2 Seconds...

play fullscreen
1 / 12

A2 – Determining Limits using Limit Laws and Algebra - PowerPoint PPT Presentation


  • 181 Views
  • Uploaded on

A2 – Determining Limits using Limit Laws and Algebra. IB Math HL&SL - Santowski. (A) Review - The Limit of a Function.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'A2 – Determining Limits using Limit Laws and Algebra' - wauna


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
a review the limit of a function
(A) Review - The Limit of a Function
  • The limit concept is the idea that as we get closer and closer to a given x value in progressively smaller increments, we get closer to a certain y value but we never quite reach this y value
  • We will also incorporate the concept of "approaching x from both sides" in our discussion of the concept of limits of a function  as we can approach a given x value either from the right of the x value or from the left
a review the limit of a function1
(A) Review - The Limit of a Function
  • ex 1. Consider a very simple function of f(x) = x² - 4x + 2 and we will be asking ourselves about the behaviour of the function near x = 2  (Set up graphing calculator to see the graph plus tables of values where we make smaller increments near 2 each time. As we do this exercise, realize that we can approach the value of x from both the left and the right sides.)
  • We can present this as lim x2 (x2 – 4x + 2) which we interpret as the fact that we found values of f(x) very close to -2 which we accomplished by considering values of x very close to (but not equal to) 2+ (meaning approaching 2 from the positive (right) side) and 2- (meaning that we can approach 2 from the negative (left) side)
  • We will notice that the value of the function at x = 2 is -2  Note that we could simply have substituted in x = 2 into the original equation to come up with the function behaviour at this point
b investigating simple limit laws
(B) Investigating Simple Limit Laws
  • With our previous example the limit at x = 2 of f(x) = x2 – x + 2 , we will break this down a bit:
  • (I) Find the following three separate limits of three separate functions (for now, let’s simply graph each separate function to find the limit)
    • lim x2 (x2) = 4
    • lim x2 (-4x) = -4 x lim x2 (x) = (-4)(2) = -8
    • lim x2 (2) = 2
  • Notice that the sum of the three individual limits was the same as the limit of the original function
  • Notice that the limit of the constant function (y = 2) is simply the same as the constant
  • Notice that the limit of the function y = -4x was simply –4 times the limit of the function y = x
c limit laws
(C) Limit Laws
  • Here is a summary of some important limits laws:
  • (a) sum/difference rule  lim [f(x) + g(x)] = lim f(x) + lim g(x)
  • (b) product rule  lim [f(x)  g(x)] = lim f(x)  lim g(x)
  • (c) quotient rule lim [f(x)  g(x)] = lim f(x)  lim g(x)
  • (d) constant multiple rule  lim [kf(x)] = k  lim f(x)
  • (e) constant rule  lim (k) = k
  • These limits laws are easy to work with, especially when we have rather straight forward polynomial functions
d limit laws examples
(D) Limit Laws - Examples
  • Find lim x2 (3x3 – 4x2 + 11x –5) using the limit laws
  • lim x2 (3x3 – 4x2 + 11x –5)
  • = 3 lim x2 (x3) – 4 lim x2 (x2) + 11 lim x2 (x) - lim x2 (5)
  • = 3(8) – 4(4) + 11(2) – 5 (using simple substitution or use GDC)
  • = 25
  • For the rational function f(x), find
  • lim x2 (2x2 – x) / (0.5x3 – x2 + 1)
  • = [2 lim x2 (x2) - lim x2 (x)] / [0.5 lim x2 (x3) - lim x2 (x2) + lim x2 (1)]
  • = (8 – 2) / (4 – 4 + 1)
  • = 6
e working with more challenging limits algebraic manipulations
(E) Working with More Challenging Limits – Algebraic Manipulations
  • But what our rational function from previously was changed slightly  f(x) = (2x2 – x) / (0.5x3 – x2) and we want lim x2 (f(x))
  • We can try our limits laws (or do a simple direct substitution of x = 2)  we get 6/0  so what does this tell us???
  • Or we can have the rational function f(x) = (x2 – 2x) / (0.5x3 – x2) where lim x2 f(x) = 0/0  so what does this tell us?
  • So, often, the direct substitution method does not work  so we need to be able to algebraically manipulate and simplify expressions to make the determination of limits easier
f evaluating limits algebraic manipulation
(F) Evaluating Limits – Algebraic Manipulation
  • Evaluate lim x2 (2x2 – 5x + 2) / (x3 – 2x2 – x + 2)
  • With direct substitution we get 0/0  ????
  • Here we will factor first (Recall factoring techniques)
  • = lim x2 (2x – 1)(x – 2) / (x2 – 1)(x – 2)
  • = lim x2 (2x – 1) / (x2 – 1)  cancel (x – 2)‘s
  • Now use limit laws or direct substitution of x = 2
  • = (2(2) – 1) / ((2)2 – 1))
  • = 3/3
  • =1
f evaluating limits algebraic manipulation1
(F) Evaluating Limits – Algebraic Manipulation
  • Evaluate
  • Strategy was to find

a common denominator

with the fractions

f evaluating limits algebraic manipulation2
(F) Evaluating Limits – Algebraic Manipulation
  • Evaluate

(we recall our earlier

work with complex numbers

and conjugates as a way

of making “terms disappear”

g internet links
(G) Internet Links
  • Limit Properties - from Paul Dawkins at Lamar University
  • Computing Limits - from Paul Dawkins at Lamar University
  • Limits Theorems from Visual Calculus
  • Exercises in Calculating Limits with solutions from UC Davis
h homework
(H) Homework
  • Stewart, 1989, Calculus – A First Course, Chap 1.2, p19, Q3-6eol, 7,8,9