Advanced Mathematics 3208 Unit 2 Limits and Continuity
Expanding • Expand the following: A) (a + b)2 B) (a + b)3
D) (x + 2)4 E) (2x -3)5
Look for Patterns A) x2 – 9 B) x3 + 27
II. Functions, Graphs, and Limits Analysis of graphs. With the aid of technology. Prelude to the use of calculus both to predict and to explain the observed local and global behaviour of a function.
Analysis of Graphs Using graphing technology: • Sketch the graph of y = x3 – 27
Analysis of Graphs 1. y = x3 – 27 A) Find the zeros B) Find the local max and min points • These are points that have either the largest, or smallest y value in a particular region, or neighbourhood on the graph. x = 3 • There are no local max or min points
C) Identify any points where concavity changes from concave up to concave down (or vice a versa). The point of inflection is (0, -27)
2. Sketch the graph of: A) B) y = x – 2 What do you notice? • y = x – 2 is a slant (or oblique) asymptote.
Rational Functions • f(x) is a rational function if where p(x) and q(x) are polynomials and • Rational functions often approach either slant or horizontal asymptotes for large (or small) values of x
Rational Functions are not continuous graphs. • There various types of discontinuities. • There vertical asymptotes which occur when only the denominator (bottom) is zero. • There are holes in the graph when there is zero/zero
3. Describe what happens to the function near x = 2. • The graph seems to approach the point (2, 4) • What occurs at x = 2? • Division by zero. The function is undefined when x = 2. In fact we get • There is a hole in the graph. • What occurs at x = -2? • Division by zero however this time there is a vertical asymptote.
4. Describe what happens to the function as x gets close to 0. • The function seems to approach 1 • Does it make any difference if the calculator is in degrees or radians? • Yes, it only approaches 1 in radians.
Limits of functions (including one-sided limits). A basic understanding of the limiting process. Estimating limits from graphs or tables of data. Calculating limits using algebra. Calculating limits at infinity and infinite limits
Zeno’s Paradox • Half of Halves • Mathematically speaking: • This is the limit of an infinite series
How many sides does a circle have? http://www.mathopenref.com/circleareaderive.html 5 sides? 18 sides?
Limit of a Function • The limit of a function tells how a function behaves near a certain x-value. • Suppose if I wanted to go to a certain place in Canada. • We would use a map
Consider: • If we have a function y = f(x) and we are trying to find out what the value of the function is for a x-value under the shaded area, we could make an estimate of what it would be by looking at the function before it goes into or leaves the shaded area. Guess what the function value is at x = 3
The smaller the shaded area can be made, the better the approximation would be. Guess what the function value is at x = 3
Mathematically speaking: • As x gets close to a, f(x) gets close to a value L • This can be written: • It means “The limit of f(x) as x approaches a equals L Note: This is not multiplication.
We can get values of f(x) to be arbitrarily close to L by looking at values of x sufficiently close to a, but not equal to a. • It does not matter if f(a) is defined. • We are only looking to see what happens to f(x) as x approaches a
Limits using a table of values. 1. Determine the behaviour of f (x) as x approaches 2.
Examples: (Using a Table of Values) 2. Find: 5 3 4.5 3.5 This is the limit from the left side of x = 2 This is the limit from the right side of x = 2 4.1 3.9 4.01 3.99 4.001 3.999
Examples: (Using a Table of Values) 2.Find: 0.998334 0.998334 0.9999833 0.9999833 0.9999998 0.9999998
3. For the function , complete the table below Sketch the graph of y = f(x)
Using the table and graph as a guide, answer following questions: • What value is f (x) approaching as x becomes a larger positive number? • What value is f (x) approaching as x becomes a larger negative number? • Will the value of f (x) ever equal zero? Explain your reasoning.
One Sided Limits This is a piecewise function Consider the function below: It consists of two different functions combined together into one function What is the equation?
Find the following using the graph and function rule A) B) C) D) For this limit we need to find both the left and right hand limits because the function has different rules on either side of 1.
= 2 = 0 • In this case we say that the limit Does Not Exist • (DNE) • NOTE: Limits do not exist if the left and right limits at a x-value are different.
Mathematically Speaking • A function will have a limit L as x approaches a, if and only if as x approaches a from the left and a from the right you get the same value, L. • OR:
2.A) Draw B) Find:
B) Find: 3.A) Draw C) Find:
Identify which limit statements are true and which are false for the graph shown.
Text Page 33-34 • 3, 4, 7, 9, 15, 18
Absolute Values • Definition: The absolute value of a, |a|, is the distance a is from zero on a number line. |3| = |-3| = |x| = 2 Note: - a is positive if a is negative
EX. |-5| • Here the value is negative so • |-5| = -(-5) = 5
5. |x| = 3 6. |x| < 3