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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