AP Calculus Notes: Limits
AP Calculus Notes: Limits – 1
Ex: [pic]
a. Natural domain:
b. What happens at x = 1?
c. What happens as x ( 1?
Note: “(” is read “approaches” or “goes to.” “x ( 1”
means x gets “arbitrarily close” to 1, from either side,
but never exactly equals 1.
1. Graph it. (Graph at right.)
2. Make a table.
[pic]
Notation: x(1– :
x(1+ :
Notation: Limit from the left:
Limit from the right:
Important notes:
1. This does not mean that f(1) = 2; f(1) is still undefined.
2. While the graph and table are both very convincing, they do not prove that the limit is 2.
Ideas of a Limit
To generalize: [pic] means
1. Informally: As x approaches a, f(x) approaches L: As x ( a, f(x) ( L.
Note: In the actual definition of a limit, x can never exactly equal a. It is possible however, that f(x) might exactly equal L in some limits.
2. Better: By getting x close enough to a (but not equal to a), we can make y as close as we like to L (but not necessarily equal to L).
Ex: Use the graphs to evaluate the following limits:
a. [pic]
b. [pic]
c. [pic]
d. [pic]
Ex: A function f is shown in the graph at right. The domain of the function is all real numbers except x = –4 and x = 0. Use the graph to evaluate the following:
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. [pic]
i. [pic]
j. [pic]
k. [pic]
AP Calculus Notes: Limits – 2
Properties of Limits
Suppose k is a constant and [pic] = L and [pic] = M . Then the following properties hold:
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. If f is continuous at M, then [pic]
Ex: If [pic], [pic] and [pic], evaluate
a. [pic]=
b. [pic]=
c. [pic]=
d. [pic]=
e. [pic]=
f. [pic]=
g. [pic]=
h. [pic]=
Evaluating Limits by Direct Substitution
Ex: Use the properties of limits to evaluate [pic]
[pic]=
Many (not all!) limits can be evaluated by “direct substitution.”
Ex: [pic]= Ex: [pic]=
Ex: [pic]=
When doesn’t it work?
Ex: [pic]
Ex: [pic]where [x] is the greatest integer function:
[x] = the greatest integer ( x.
Ex: [pic] and [pic]where [pic]
AP Calculus Notes: Limits – 3
Evaluating Limits Algebraically
If a function f is continuous at x = a (there is no break in the graph of f at x = a), then the limit [pic]can be evaluated by direct substitution: [pic]. Direct substitution does not work if f(a) evaluates to [pic]. It does not necessarily mean the limit DNE. It does mean we need to do extra analysis (work).
Our strategy will be to try to show that f is identical, except at x = a, to some other function g for which direct substitution does work.
Ex: a) [pic]
b) [pic]
c) [pic]
d) [pic]
e) [pic]
f) [pic]
g) [pic]
Summary
To evaluate a limit:
1. Try direct substitution. Unfortunately, this does not work for most of the important limits in calc.
2. If the function is piecewise defined with a break at the limit, evaluate the limit on both sides. See if the one-sided limits are the same. (See last example from Limits – 2).
3. If direct substitution gives [pic] where N (0, then the limit DNE (can’t be “fixed”) (ex e above)
4. If direct substitution gives [pic] then try appropriate algebraic “tricks:”
a. factoring and reducing (ex a, e above)
b. rationalizing numerator or denominator (ex c, d)
c. simplifying complex fractions (ex b)
d. multiplying out and simplifying (ex g)
5. If all else fails, look at graph and/or table. (Remember, these are good for exploring or checking but are not considered “proofs.”)
AP Calculus Notes: Limits – 5
Continuity
Informal definition: A function is continuous at a point x = c if the graph of the function does not have a break there. A function is continuous on an interval (a, b) if it is continuous at each point in the interval; in other words, if it has no breaks in the interval.
Ex: a) [pic]
b) [pic]
c) [pic]
d) [pic]
e) [pic]
Continuity: Formal Definition
A function f is continuous at the point x = c if
1. f(c) is defined
2. [pic] exists
3. [pic]
Ex: b) [pic]
c) [pic]
e) [pic]
Ex: Find the value of a that will make f continuous at x = 2 if [pic].
AP Calculus Notes: Limits – 6
Intermediate Value Theorem
Ex: You go for a hike in the Adirondack Mountains. At 8:00 AM you are at 1500’ elevation; at noon you are at 3500’ elevation. Between 8:00 AM and noon, can you say for sure if you were ever at an elevation of
a. 2500'?
b. 4000'?
Intermediate Value Theorem (IVT): If f(x) is continuous on [a, b] and N is any value strictly between f(a) and f(b), then there is at least one value c in (a, b) such that f(c) = N.
In the Adirondack hiking example above,
[a, b] = f(a) = f(b) = N =
What (reasonable) assumption did we make about our elevation as a function of time?
Ex: Does [pic] have a root in [1, 2]?
Ex: A continuous function g has domain 0 ( x ( 10. Certain values of g are shown in the table below.
What is the least possible number of solutions to the equation g(x) = 7 if
a. k = 6
b. k = 7
c. k = 8
Ex: Does [pic] = 1 somewhere in the interval [1, 3]?
AP Calculus Notes: Limits – 7
Infinite Limits
Ex: Describe the behavior of [pic] near x = 3.
Notes:
1. [pic] DNE. Saying it “equals (” is simply telling more precisely why it DNE
2. [pic] means “As x approaches c, f(x) gets larger and larger without bound (with no limit).”
Ex: Ex: Describe the behavior of [pic] near x = 3.
Vertical Asymptotes
If [pic] or ([pic], or if either of the one-sided limits at x = c equals [pic] or ([pic], then f has a vertical asymptote at x = c.
Ex: [pic]
Ex: [pic]
Ex: [pic]
AP Calculus Notes: Limits – 8
Limits at Infinity
Ex: Evaluate [pic].
The following facts are helpful when evaluating limits at ((:
1. For large x, a polynomial function behaves like its highest order term.
2. An exponential function ax (a > 1) grows faster than any power of x.
3. Any positive power of x grows faster than a log function logax (a > 1).
Ex: Evaluate the following
a. [pic] b. [pic]
c. [pic] d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. [pic]
i. [pic]
Horizontal Asymptotes
If [pic] or [pic], then f has a horizontal asymptote at y = b.
Ex: a. [pic]
b. [pic]
c. [pic]
d. [pic]
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(
[pic]
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f(a)
a
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[pic]
[pic]
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|x |0.9 |0.99 |0.999 |0.9999 |x ( 1 |1.0001 |
|g(x) |11 |9 |3 |5 |k |4 |
20
10
0
-10
-20
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-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
x
1
-1
1 2
x
x
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