Download Math 111 Week Number Four Notes

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Math 111 – Calculus I.
Week Number Four Notes
Fall 2003
I.
Continuity of a function of a real variable
Now that we understand limits of functions at a real number a, we can discuss the notion
of a function being continuous at a value a. The mathematical notion of continuity
closely mirrors the intuitive notion of what it would mean for a function to be continuous
near a value a. We can “draw” the function near a without “lifting the pencil” (i.e. the
function has no “breaks” or “jumps” near a. Let’s now be more rigorous about our
definition.
Definition 4.1: Assume f is a real-valued function with domain a subset of the real line.
Assume a is in the domain of f. Then, f is continuous at a if
lim f(x)  f(a)
x a
What does the above definition mean intuitively? What makes this different from the
definition of f having a limit at the value a. Here is my “intuitive explanation” of what
this definition means.
As x “gets arbitrarily close to a”, f(x) “gets arbitrarily close to f(a)” (note that this implies
that f(a) EXISTS).
EXERCISE: Give analogous definitions for left and right continuity, respectively, of a
function f at a real number a.
First let’s look at sketches of some functions to understand how this concept works.
Example 4.2: Consider the sketches of the following functions f(x), g(x), and h(x) below.
f(x)
a
g(x)
h(x)
a
a
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Does the limit as x approaches a of f(x) exist? Is f continuous at x = a? If so, what is it?
If not, explain why not?
Does the limit as x approaches a of g(x) exist? Is g continuous at x = a? If so, what is it?
If not, explain why not?
Does the limit as x approaches a of h(x) exist? Is h continuous at x = a? If so, what is it?
If not, explain why not?
Now that we have an idea on how to estimate limits of functions graphically, we will
focus on determining the continuity of functions defined via some form of formula. Let’s
look at the following example.
Example 4.3:
For each of the functions in parts (a) and (b) below, determine the REAL NUMBERS x
for which the function is discontinuous. JUSTIFY YOUR ANSWER.
(a ) f(x) 
x 2  9x  8
x2
(b) g(x)  x - x where x represents the greatest integer  x.
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The following example will illustrate how we can glue continuous functions together
“piece by piece” to create a continuous function on a larger domain.
Example 4.4: Find constants C and K that make the following function f defined below
continuous on the entire real line. Sketch the graph of f(x).
Kx 2
if x  /2
2cos(x) if /2  x  3/2
f(x) =
Lx  3 if x  3/2
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II.
Some rules for constructing continuous functions and classes of
continuous functions
The following two results will show us how to construct continuous functions from other
continuous functions and also tell us that many of the functions that we will be studying
in this course are continuous on their domains.
Theorem 4.5: Assume that f and g are continuous functions at a real number a, c is a
constant and n is a positive integer. Then, the following functions are continuous at x =
a.
(i) f + g
(ii) f – g
(iii) cf
(v) f/g if g(a) is not 0
(iv) fg
(vi) fn
(vii) f1/n is continuous at a if n is odd. If n is even, f1/n is continuous at a assuming a >
0.
Finally, (viii) if h is continuous at f(a), then h f is continuous at a.
Theorem 4.6: The following classes of functions are continuous on their implied
domains (as maximal subsets of the real line).
(i) polynomials (ii) rational functions (iii) root functions (iv) trigonometric functions
(v) inverse trigonometric functions (vi) exponential functions
(vii) logarithmic functions
Finally, the following theorem is the basis for developing algorithms that computers can
use to display graphs of continuous functions. It is called the intermediate value theorem.
Theorem 4.7(The Intermediate Value Theorem): Assume f is continuous on the closed
interval [a,b]. Assume R is any real number between f(a) and f(b) (excluding the
endpoints). Then, there exists a c in (a,b) such that f(c) = R.
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Intuitively, this says that all values between f(a) and f(b) are in the image of the function f
(i.e. they are “hit by f”).
Here is an example of an application of this theorem.
Example 4.8(problem 33, p. 129 of Stewart): Assume that f(x) = x3 – x2 + x, show that
there is a number c such that f(c) = 10.
III.
Unbounded Limits and Limits at Infinity
We will begin by looking at unbounded limits. When the value of a function is growing
without bound as x approaches a real number a, this usually indicates the existence of a
vertical asymptote at the value a which provides “global” information about the function
on a neighborhood of a. Let’s formalize this notion with some terminology.
Definition 4.9: The following notation
lim f(x)  
x a
indicates that the value of f(x) becomes “arbitrarily large” as x approaches a. More
precisely, for every real number R, there exists a  such that for any x such that 0< |xa| < f(x) > R. We read the aforementioned notation above as follows: the limit as x
approaches a of f(x) is positive infinity.
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EXERCISE: Design analogous notation and terminology for each of the following:
(1) The limit as x approaches a of f(x) is negative infinity.
(2) Analogues of the above definition for left-hand limits and right-hand limits,
respectively.
Let’s look at some pictures of some graphs below and see if we understand this concept
intuitively.
Consider the graphs of the following two functions below.
f(x) = tan(x) on (-3g(x) = 1/x2 on (-2,2)
Answer the following questions. What is
lim f(x)?
x   / 2
lim f(x)?
x   / 2 
lim f(x)? lim g(x)?
x  /2-
x 0
This motivates the definition of a vertical asymptote. More precisely, the line x = a if a
vertical asymptote of f if f has an unbounded limit, an unbounded left-hand limit, or an
unbounded right-hand limit at x = a.
Hence, in the above examples, x = and x = - are vertical asymptotes of f and x = 0
is a vertical asymptote of g. Are there any others in the above example?
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To find vertical asymptotes of functions, we search for real numbers that are not in the
domain of the function where we might suspect there would be an unbounded limit as the
x-value approaches the given number. For rational functions, we search for roots of the
polynomial in the denominator of the function as POTENTIAL LOCATIONS of
asymptotes. The problem becomes more complicated for other classes of functions.
Let’s do one last example.
Example 4.10: Find all vertical asymptotes (if any exist) of the functions below over the
specified interval.
(a) f(x)  csc(2x) for x in (-/2, /2)
(b) g(x) 
x 2  7x  6
x 2  6x
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We’ll study limits at infinity and horizontal asymptotes at the beginning of the next note
set.
Non Hand-In Homework Problems Associated with Week #4 Notes
Sections 2.4 and 2.5
Section 2.4: 3-7,10,11,13-16, 17,20,22, 25-28, 29,31,32,34,38,46
Section 2.5: 1 (b) and (c), 4 (c) – (e), 10, 13-17,35,36, 42
Read Sections 2.6 through 2.8 of the textbook
EXAM #1: Friday, September 19th -- Pre-Calculus Review material plus Sections 2.1-2.4