LIMITS Name: _________________________________________________________ Mrs. Upham 2019-2020

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LIMITS

Name: _________________________________________________________

Mrs. Upham

2019-2020

Lesson 1: Finding Limits Graphically and Numerically

When finding limits, you are finding the y-value for what the function is approaching. This can be done in three ways:

1. Make a table

2. Draw a graph

3. Use algebra

Limits can fail to exist in three situations:

1.

2. The left-limit is different than the right-side limit.

3. Unbounded Behavior

4. Oscillating Behavior

Verbally: If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, then the limit of f(x) as x approaches c is L.

Graphically:Analytically:

Numerically: From the table,

x

-5.01

-5.001

-5

-4.999

-4.99

f(x)

3.396

3.399

3.4

3.398

3.395

1. Use the graph of f(x) to the right to find

2. Use the table below to find

x

1.99

1.999

2

2.001

2.01

f(x)

6.99

6.998

ERROR

7.001

7.01

3. Using the graph of H(x), which statement is not true?

a.

b.

c. does not exist

d.

Lesson 2: Finding Limits Analytically

Properties of Limits

Some Basic Limits

Let b and c be real numbers and let n be a positive integer.

Methods to Analyze Limits

1. Direct substitution

2. Factor, cancellation technique

3. The conjugate method, rationalize the numerator

4. Use special trig limits of or

Direct Substitution

1.

2.

3.

4.

5.

6.

7. If then

8.

9.

10. Given: and , find:

a.

b.

c.

Limits of Polynomial and Rational Functions:

11.

12.

13.

14.

Limits of Functions Involving a Radical

15.

Dividing out Technique

16.

17.

18. Given f(x) = 3x + 2

Find

Special Trigonometric Limits:

19.

20.

21.

22.

23.

24.

25.

The Squeeze Theorem

If h(x) < f(x) < g(x) for all x in an open interval containing c, except possibly at c itself, and if then exists and is equal to L.

4 – < f(x) < 4 +

Lesson 3: Continuity and One-Sided Limits

Definition of Continuity

Continuity at a point:

A function f is continuous at c if the following three conditions are met:

1. f(c) is defined

2. exists

3.

Properties of continuity:

Given functions f and g are continuous at x = c, then the following functions are also continuous at x = c.

1. Scalar multiple:

2. Sum or difference: f± g

3. Product: f • g

4. Quotient: , if g(c) ≠ 0

5. Compositions: If g is continuous at c and f is continuous at g©, then the composite function is continuous at c,

The existence of a Limit:

The existence of f(x) as x approaches c is L if and only if and

Definition of Continuity on a Closed Interval:

A function f is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and and

Example 3: Given for what values of x is h not continuous? Justify.

Example 4: If the function f is continuous and if f(x) = when x ≠ -2, then f(-2) = ?

Example 5: Which of the following functions are continuous for all real numbers x?

a. y =

b. y = ex

c. y = tan x

A) NoneB) I onlyC) II onlyD) I and III

Example 6: For what value(s) of the constant c is the function g continuous over all the Reals?

Lesson 4: The Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is an existence theorem which says that a continuous function on an interval cannot skip values. The IVT states that if these three conditions hold, then there is at least one number c in [a, b] so that f(c) = k.

1. f is continuous on the closed interval [a, b]

2. f(a) ≠ f(b)

3. k is any number between f(a) and f(b)

Example 1: Use the Intermediate Value Theorem to show that f(x) = + 2x – 1 has a zero in the interval [0, 1].

Example 2: Apply the IVT, if possible, on [0, 5] so that f(c) = 1 for the function f(x) =

Example 3: A car travels on a straight track. During the time interval 0 < t < 60 seconds, the car’s velocity v, measured in feet per second is a continuous function. The table below shows selected values of the function.

t, in seconds

0

15

25

30

35

50

60

v(t) in ft/sec

-20

-30

-20

-14

-10

0

10

A. For 0 < t < 60, must there be a time t when v(t) = -5?

B. Justify your answer.

Example 4: Find the value of c guaranteed by the Intermediate Value Theorem.

f(x) = x2 + 4x – 13 [0, 4] such that f(c) = 8

Lesson 5: Infinite Limits

Definition of Vertical Asymptotes:

A vertical line x = a is a vertical asymptote if and/or has a vertical asymptote at x = c.

Properties of Infinite Limits:

Let c and L be real numbers and let f and g be functions such that and

1. Sums or Difference:

2. Product:

3. Quotient:

Example 1: Evaluate by completing the table for

x

-3.5

-3.1

-3.01

-3.001

-3

-2.999

-2.99

-2.9

-2.5

f(x)

Example 2: Evaluate

Example 3: Evaluate

Example 4: Evaluate

Example 5: Evaluate

Example 6: Evaluate

Example 7: Evaluate

Example 8: Find any vertical asymptotes or removable discontinuities

Example 9: Determine whether the graph of the function has a vertical asymptote or a removable discontinuity at x = 1. Graph the function to confirm

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