MM-Limits and Continuity
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(Chapter 2: Limits and Continuity) 2.0.1
CHAPTER 2: LIMITS AND CONTINUITY
2.1: An Introduction to Limits 2.2: Properties of Limits 2.3: Limits and Infinity I 2.4: Limits and Infinity II 2.5: The Indeterminate Forms 0/0 and / 2.6: The Squeeze (Sandwich) Theorem 2.7: Precise Definitions of Limits 2.8: Continuity
In Swokowski (Classic / 5th ed.) 2.1, 2.3 2.3 2.4 2.4 2.1, 2.3, 2.4 2.3 2.2 2.5
In Thomas (11th ed.) 2.1, 2.4 2.2 2.4 2.5 2.1, 2.2, 2.4, 2.5 2.2 2.3 2.6
ASSUMPTIONS THROUGHOUT THE NOTES Unless otherwise specified … • We assume that f and g denote functions. • We assume that a, b, c, and k denote real constants. • We assume that the domain of a function is its implied domain. • We assume that graphs extend beyond the scope of the figures in an expected manner, unless endpoints are clearly shown. Arrowheads may help to make this clearer. • Before we get to multivariable calculus, we will assume that “real constants” are “real constant scalars,” as opposed to vectors.
(Section 2.1: An Introduction to Limits) 2.1.1
SECTION 2.1: AN INTRODUCTION TO LIMITS PART A: “EASY” EXAMPLES Example 1 (Polynomial Function)
()
()
Assuming f x = 3x 2 + x 1 , evaluate lim f x . x1
What is this asking? means “approaches.” We will discuss this more rigorously later. lim f x is read: “the limit of f x as x approaches 1.” x1
()
()
()
It is the real number that f x approaches as x approaches 1, if such a number exists. Solution Method f is a polynomial function with presumably unrestricted domain R. Here, we substitute (“plug in”) x = 1 and evaluate f 1 .
()
Warning 1: Sometimes, lim f x does not equal x a
() f ( a ) . We will see
examples of this later. Solution
()
(
)
lim f x = lim 3x 2 + x 1
x1
x1
Warning 2: When taking the limit of an expression consisting of more than one term, make sure to group the entire expression.
() () 2
= 3 1 + 1 1 Warning 3: When performing substitutions, be prepared to use grouping symbols unless you are sure that they are not required. =3
(Section 2.1: An Introduction to Limits) 2.1.2 Notation We can write: lim f x = 3 . x1
()
()
Alternatively, we can write: f x 3 as x 1. A graph can demonstrate this. Consider the graph of y = f x ; here, y = 3x 2 + x 1.
()
(Figure 2.1.a) Imagine that the arrows in the figure above represent two lovers running towards each other along the parabola. What is the y-coordinate of the point that they are approaching? It is 3. (Remember that y-coordinates correspond to function values here.)
Example 2 (Rational Function)
()
Assuming f x =
2x + 1 , evaluate lim f x . x3 x2
()
Solution Method f is a rational function with implied domain x R x 2 .
{
}
Here, we observe that 3 is in the domain of f , so we substitute (“plug in”) x = 3 and evaluate f 3 .
()
(Section 2.1: An Introduction to Limits) 2.1.3 Solution
()
2x + 1 x3 x 2 2 3 +1
lim f x = lim
x3
=
() (3) 2
=7 A graph can demonstrate this.
()
Consider the graph of y = f x ; here, y =
2x + 1 . x2
(Figure 2.1.b) Note: You might not know how to graph the entire graph until later. This will often be the case with these figures. But wait! What if the lover on the left is running along the left branch of the graph? The left branch is irrelevant to our analysis. We really only care about what happens when the lovers are in the “immediate vicinity” of x = 3; this will be explained rigorously later. LIMITS ARE “LOCAL.”
By the way, what are the brown dashed lines? They’re called vertical and horizontal asymptotes (VAs and HAs), which are lines that the graph approaches in a “long-run” or “explosive” sense. We’ll define them using limits later on.
(Section 2.1: An Introduction to Limits) 2.1.4 Example 3 (Constant Function)
lim 2 = 2 .
x
A graph can demonstrate this. Think: f x = 2 . Consider the graph of y = 2 .
()
(Figure 2.1.c) The term “approaches” We can write 2 2 (i.e, 2 approaches 2) as x . This is true, even though people are more used to thinking about a sequence of distinct numbers such as 2.1, 2.01, 2.001, … approaching 2. It helps to remember that, in math, the constant sequence 2, 2, 2, … is also said to approach 2. Remember that all constant functions are also polynomial functions, and all polynomial functions are also rational functions.
A Limit Theorem for Rational Functions If: f is a rational function; its domain, Dom f , is its implied domain; and
( )
( )
a is a real constant in Dom f , then:
()
()
lim f x = f a .
x a
That is, to compute the limit, substitute (“plug in”) x = a and evaluate f a .
()
We will justify this theorem later.
(Section 2.1: An Introduction to Limits) 2.1.5 Be prepared to work with function and variable names other than f and x. Example 4 (Revisiting Example 2)
()
Assuming g t =
2t + 1 , evaluate lim g t . t3 t2
()
Solution
()
2t + 1 t3 t 2 2 3 +1
lim g t = lim
t3
=
() (3) 2
=7 PART B: ONE- AND TWO-SIDED LIMITS; EXISTENCE OF LIMITS
()
The limit problems of the form lim f x in Part A were two-sided limit problems, x a
because we were interested in what happened when we approached x = a from both sides: from the left and from the right. One-sided limit problems only focus on one of these approaches. Example 5 (Left-Hand Limit)
(
)
Evaluate lim x + 3 . x3
What is this asking? This is an example of a left-hand limit problem, which is a type of one-sided limit problem.
()
()
Let f x = x + 3 . lim f x is read:
()
x3
“the limit of f x as x approaches 3 from the left.”
()
It is the real number that f x approaches as x approaches 3 from lesser or lower numbers (imagine approaching x = 3 from the left along the real number line), if such a number exists.
(Section 2.1: An Introduction to Limits) 2.1.6 Solution Method We use the same method that we used in Part A for two-sided limits. f is a rational (in fact, polynomial) function with implied domain R. Here, we observe that 3 is in the domain of f, so we substitute (“plug in”) x = 3 and evaluate f 3 .
()
Solution
lim
x3
( x + 3) = 3 + 3 =6
Numerical / Tabular Method Although it is not a comprehensively convincing method that is typically accepted on exams, and it can even be misleading at times, this method is at least easy to understand. Select an increasing sequence of real numbers that approaches 3 such that all the numbers are less than 3. Then, evaluate the function at those numbers and take a guess as to what number, if any, the function values are approaching. For example:
()
f x = x+3
x 2.9 2.99 2.999
5.9 5.99 5.999
6 (?)
3 or x f x = x+3
()
(
2.9
2.99
2.999
5.9
5.99
5.999
3 6 (?)
)
We guess: lim x + 3 = 6 . x3
Warning 4: Be careful about associating the “ ” superscript with negative numbers. Here, when considering the left-hand limit, we want to consider positive numbers that are close to (but less than) 3. If we were taking a limit as x approached 0, then we would associate the “ ” superscript with negative numbers and the “+” superscript with positive numbers.
(Section 2.1: An Introduction to Limits) 2.1.7 A graph can demonstrate this. Consider the graph of y = f x ; here, y = x + 3 .
()
(Figure 2.1.d) We only care about the lover to the left of x = 3. (A stand-up, perhaps?)
Example 6 (Right-Hand Limit)
(
)
Evaluate lim+ x + 3 . x3
What is this asking? This is an example of a right-hand limit problem, which is a type of one-sided limit problem.
()
()
Let f x = x + 3 . lim+ f x is read:
()
x3
“the limit of f x as x approaches 3 from the right.”
()
It is the real number that f x approaches as x approaches 3 from greater or higher numbers (imagine approaching x = 3 from the right along the real number line), if such a number exists. Solution
lim
x3 +
( x + 3) = 3 + 3 =6
(Section 2.1: An Introduction to Limits) 2.1.8 Numerical / Tabular Method Select a decreasing sequence of real numbers that approaches 3 such that all the numbers are greater than 3. Then, evaluate the function at those numbers and take a guess as to what number, if any, the function values are approaching. For example: x 3.1 3.01 3.001
()
f x = x+3 6.1 6.01 6.001
6 (?)
3 or x 3 f x = x + 3 6 (?)
()
(
3.001 6.001
3.01 6.01
)
We guess: lim+ x + 3 = 6 . x3
A graph can demonstrate this. Consider the graph of y = f x ; here, y = x + 3 .
()
(Figure 2.1.e) We only care about the lover to the right of x = 3. (Maybe the lovers have bad timing?)
3.1 6.1
(Section 2.1: An Introduction to Limits) 2.1.9
Existence of Limits A limit exists (if and only if, or iff) the limit can be expressed as a single real constant. Otherwise, the limit does not exist (“DNE”). Later, we will be able to say that a limit is (infinity) or (negative infinity) in some cases, but the limit is still nonexistent in those cases. The notation in those cases indicates why the limit does not exist.
Two-Sided Limits If a and L are real constants, then lim f x = L x a
()
() lim f ( x ) = L ). x a +
( lim f x = L , and x a
That is, a two-sided limit exists the left-hand and right-hand limits exist, and they equal the same real constant. The value of the two-sided limit then equals that constant. If either one-sided limit does not exist (DNE), or if the two one-sided limits exist but are unequal, then the two-sided limit does not exist (DNE).
Example 7 (Revisiting Examples 5 and 6)
( x + 3) = 6 , and lim ( x + 3) = 6 , so x3 + lim ( x + 3) = 6 . x3 lim
x3
(Section 2.1: An Introduction to Limits) 2.1.10
Revisiting our Limit Theorem for Rational Functions If: f is a rational function; its domain, Dom f , is its implied domain; and
( )
( )
a is a real constant in Dom f , then:
() () lim f ( x ) = f ( a ) , and, therefore, x a + lim f ( x ) = f ( a ) . x a lim f x = f a ,
x a
That is, to compute these limits, substitute (“plug in”) x = a and evaluate f a .
()
Let’s look at a function that is not rational. The aforementioned theorem will not apply. Example 8
()
Let f x = x + 1 .
( ) {
}
)
Observe that Dom f = x R x 0 , or 0, .
(Figure 2.1.f)
(Section 2.1: An Introduction to Limits) 2.1.11
()
Right-Hand Limit: lim+ f x = 1. x0
This is because the function values approach 1 as we approach x = 0 from the right.
()
Left-Hand Limit: lim f x does not exist (DNE). x0
This is because function values in this example are undefined as we approach x = 0 from the left. In order for the left-hand limit to exist, there must exist an interval of the form c, 0 on
( )
(i.e., throughout) which the function is defined, for some negative real constant c. That is not the case here.
( )
By the way, why did we give the interval form c, 0 and not
( c, 0 above? We do not require the function to be defined at 0, itself! We will discuss this issue later.
()
Two-Sided Limit: lim f x does not exist (DNE). x0
This is because the corresponding left-hand limit does not exist (DNE). Observe that f is not a rational function, so the aforementioned (revised) Limit Theorem for Rational Functions does not apply, even though 0 is in Dom f . f is, however, an algebraic function, and we will discuss a limit
( )
theorem for algebraic functions later.
(Section 2.1: An Introduction to Limits) 2.1.12 PART C: THERE DOESN’T HAVE TO BE A “POINT”! “IGNORE a” THEOREMS Example 9 (Modifying Examples 5-7)
()
(
)
Let g x = x + 3, x 3 . We are removing 3 from the domain of the function from Examples 5-7.
(Figure 2.1.g)
( )
The point 3, 6 is no longer on the graph. Instead, we have a hole; later, we will say that there is a removable discontinuity at x = 3.
()
Nevertheless, even though f 3 is now undefined, the following statements are true:
() lim g ( x ) = 6 , and x3 + lim g ( x ) = 6 . x3 lim g x = 6 ,
x3
In Examples 5-7, the limit value was attained by the function at x = 3. Here, it is not!
(Section 2.1: An Introduction to Limits) 2.1.13
()
()
f a may or may not be relevant to lim f x x a
()
()
The existence of lim f x does not require the existence of f a . x a
(See Example 9.)
()
()
Even if f a exists, lim f x could be a different value, or it might not x a
exist at all. (See Example 10.)
()
()
If lim f x = f a , then f is continuous at a, as it was in x a
Examples 5-7 for a = 3; we will discuss continuity later. Example 10 (Modifying Example 9)
x + 3, x 3 Let the function h be defined piecewise as follows: h x = x=3 7,
()
(A piecewise-defined function uses different evaluation rules for different subsets of – i.e., different groups of values in – its domain. This type of function can lead to interesting limit problems.)
()
Evaluate lim h x . x3
What does the graph of h suggest?
(Figure 2.1.h)
()
()
lim h x = 6 once again, even though h 3 = 7 .
x3
(Section 2.1: An Introduction to Limits) 2.1.14 Suggested solution
()
(
)
lim h x = lim x + 3
x3
x3
= 3+ 3 =6 Why is the suggested solution appropriate? We only care about the behavior of the h function in the “immediate vicinity” of x = 3, excluding x = 3, itself.
()
The function rule h x = x + 3 applies to the values of x that are in the “immediate vicinity” of x = 3, excluding x = 3, itself. More precisely, we can find an open interval containing 3, say 2.9, 3.1 or even the entirety of R, on which the h function is defined
(
)
()
using the function rule h x = x + 3 , except at x = 3, itself.
()
Therefore, h x = x + 3 is the only rule that is relevant when we consider approaching x = 3 from the left or from the right.
()
(
)
As a consequence, either lim h x = lim x + 3 , or neither limit
(
x3
)
x3
exists. We know lim x + 3 = 6 , so we can conclude that
()
lim h x = 6.
x3
x3
(Section 2.1: An Introduction to Limits) 2.1.15 How do we generalize this approach? (Perhaps look at Example 11 now.)
The "Ignore a" Theorem for Two-Sided Limits:
()
Evaluating the two-sided limit lim f x x a
even if f is not a rational function with a in its domain If: f is a function that is defined by the function rule r x on (i.e., throughout) some open x-interval
()
containing the real constant a, possibly excluding a, itself, then:
()
()
lim f x = lim r x , or neither limit exists.
x a
x a
We can develop modified theorems for one-sided limits as follows. These modifications will be made clearer in Example 11. Basically, when evaluating a left-hand limit, we use the function rule that governs the x-values “immediately to the left” of a on the real number line. Likewise, when evaluating a right-hand limit, we use the rule that governs the x-values “immediately to the right” of a.
The "Ignore a" Theorem for Left-Hand Limits If: f is a function that is defined by the function rule r x on (i.e., throughout) some open x-interval of
()
( )
the form c, a , where c is a real constant and c < a , then:
()
()
lim f x = lim r x , or neither limit exists.
x a
x a
The "Ignore a" Theorem for Right-Hand Limits If: f is a function that is defined by the function rule r x on (i.e., throughout) some open x-interval of
()
( )
the form a, c , where c is a real constant and c > a , then:
()
()
lim f x = lim+ r x , or neither limit exists.
x a +
x a
(Section 2.1: An Introduction to Limits) 2.1.16 What’s the bottom line? Does it matter what happens to a function f at a or not when we evaluate limits like lim f x ? x a
()
In theory, we’re not supposed to care what happens to the function at x = a . Often, though, it helps to know what happens at x = a . For example, if we have a rational function f that has a in its domain (and the domain is the implied domain), then lim f x = f a . (We will come back to this issue when we cover x a
()
()
continuity.)
Example 11 Let the function f be defined piecewise as follows:
3, if x 0 f x = 2x 2 , if 0 < x < 1 2x, if x > 1
()
()
Consider the graph of y = f x .
(Figure 2.1.i)
(Section 2.1: An Introduction to Limits) 2.1.17
()
lim f x = lim 2x 2
x1
x1
()
=2 1
2
=2
()
x1
()
=2 1 =2
()
()
lim f x = lim 3
x0
( )
(
)
The right-hand limit as x 1 + : The relevant function rule is 2x , because that rule applies to the x-values in an open interval of the form 1, c , where c > 1;
( )
(
)
for example, consider the interval 1, 1.1 .
lim f x = 2
x1
The relevant function rule is 2x 2 , because that rule applies to the x-values in an open interval of the form c, 1 , where c < 1; for example, consider the interval 0.9, 1 .
lim f x = lim+ 2x
x1+
The left-hand limit as x 1 :
x0
=3
The two-sided limit as x 1 : The left-hand and right-hand limits as x 1 exist and are equal, so the two-sided limit exists and equals their common value. The left-hand limit as x 0 : The relevant function rule is 3, because that rule applies to the x-values in an open interval of the form c, 0 , where c < 0 ; for
( )
(
)
example, consider the interval 0.1, 0 .
()
lim+ f x = lim+ 2x 2
x0
x0
()
=2 0
2
=0
()
lim f x
x0
does not exist (DNE)
The right-hand limit as x 0 + : The relevant function rule is 2x 2 , because that rule applies to the x-values in an open interval of the form 0, c , where c > 0 ;
( )
(
)
for example, consider the interval 0, 0.1 .
The two-sided limit as x 0 : The left-hand and right-hand limits as x 0 exist but are unequal, so the two-sided limit does not exist (DNE).
(Section 2.1: An Introduction to Limits) 2.1.18 PART D: MORE EXAMPLES OF LIMITS THAT DO NOT EXIST (DNE) Example 11
1 Let f x = sin . x
()
()
()
()
Evaluate lim f x , lim+ f x , and lim f x . x0
x0
x0
(Figure 2.1.j) As x approaches 0 from the left or from the right, the function values oscillate between 1 and 1. They do not approach a single real constant as x approaches 0 from the left, nor from the right. Therefore,
() lim f ( x ) does not exist (DNE), and x0 + lim f ( x ) does not exist (DNE). x0 lim f x does not exist (DNE),
x0
()
We say that we have “evaluated” lim f x , even though the limit x0
does not exist (DNE) and has no real value. Note: The y-axis is not a vertical asymptote (VA) here, because the graph and the function values are not “exploding” around the y-axis.
(Section 2.1: An Introduction to Limits) 2.1.19 Example 12
()
Let f x =
1 . x
(Figure 2.1.k) When we discuss (infinity) and (negative infinity) in a later section, we will be able to say:
() lim f ( x ) = , and x0 + lim f ( x ) does not exist (DNE). x0 lim f x = ,
x0
In fact, all three indicated limits do not exist; the first two statements indicate why those limits do not exist.
()
lim f x , for example, does not exist, because the function values
x0
do not approach a single real constant as x approaches 0 from the left. We will revisit this function in later sections.
(Section 2.1: An Introduction to Limits) 2.1.20 Example 13
()
Let f x =
x x
.
Note: f is not a rational function, but it is an algebraic function, since x x2 f x = = . x x
()
Remember the piecewise definition of x :
x, if x 0 x = x, if x < 0
x if x > 0 x = 1, Then, f x = = x x = 1, if x < 0 x and f 0 is undefined.
()
x
()
(Figure 2.1.letter l)
() lim f ( x ) = 1, and x0 + lim f ( x ) does not exist (DNE) x0 lim f x = 1 ,
x0
due to the fact that the aforementioned left-hand and right-hand limits exist but are unequal.
(Section 2.1: An Introduction to Limits) 2.1.21. FOOTNOTES 1. Limits do not require continuity. In a later section, we will discuss continuity, a property of many functions that helps the lovers run along the graph of a function without having to jump or hop. In the first few problems of this section, we had the luxury of imagining the lovers running towards each other (one from the left, one from the right) while staying on the graph of f and without having to jump or hop, provided they were placed on appropriate parts of the graph. Sometimes, the “run” really requires jumping or hopping. For example, consider the following function f. It turns out to be true that lim f x = 0 . x0
()
0, if x is a rational value f x = x, if x is an irrational value
()
2. Misconceptions about limits. See “Why Is the Limit Concept So Difficult for Students?” by Sally Jacobs in the Fall 2002 edition (vol.24, No.1) of The AMATYC Review, pp.25-34. • Students can be misled by the use of the word “limit” in real-world contexts. For example, a speed limit is a bound that is not supposed to be exceeded; there is no such restriction on limits in calculus. • Limit values can sometimes be attained. For example, if a function f is continuous at x = a (see Examples 5-7 for a = 3 ), then the function value takes on the limit value at x = a. • Limit values do not have to be attained. See Examples 9 and 10. Observations: • The dynamic view of limits, which involve ideas of motion and approaching (for example, our lovers), may be more accessible to students than the static view preferred by many textbook authors. The static view is exemplified by the formal definitions of limits we will see later. The dynamic view greatly assists students in transitioning to the static view and the formal definitions. • Leading mathematicians in 18th- and 19th-century Europe had heated debates about ideas of limits. 3. Multivariable calculus. When we go to higher dimensions, there will be more than two possible approaches when dealing with limit problems! 4. An example where a left-hand limit exists but not the right-hand limit.
1 if x < 0 x sin , x + x 1+ x x 1 Let g x = . sin = x x 1 2 + x sin , if x > 0 x
(
()
)
(
)
()
(Figure 2.1.m)
Then, lim f x = 0 , which can be proven by the Squeeze (Sandwich) Theorem, something x 0
()
we will cover in a later section. However, lim+ f x does not exist (DNE). x 0
See William F. Trench, Introduction to Real Analysis (free online), p.39.
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