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Warmup (no calculator):

2.1a- Limits (Day 1)

http://www.online.math.uh.edu/HoustonACT/

“The limit of fof x as x approaches c is L.”

The limit exists if the function approaches the same

single number (L) from both sides of c

Limit notation:

Read as:

read as “the limit of f(x) as x approaches 1 is 3”

Reason 2

Reason 3 (sin 1/x)

Reason 1

5 Ways to EVALUATE A LIMIT:

#1: Graphically:

(visually)

Graph the function and see where it’s going

#2: Numerically:

(table)

Make a table of values very close to the

value you want to evaluate …

(be sure to use numbers to the right AND

to the left whenever possible)

#3: Substitution:

Just plug in the value where you want to

evaluate the limit.

If you get … DO SOMETHING ELSE!

5 Ways to EVALUATE A LIMIT (con’t):

#4: Analytically (Algebraically):

- -Factor the numerator and denominator and cancel any like factors.
- Multiply by a factor of 1 (Includes rationalizing the numerator,denominator)
- - Simplify the equation use algebraic properties and/or trigonometric identities.

#5: Sandwich Theorem:

Find two other functions that bound the original

function AND that go to the same place at the point you are

trying to evaluate the limit.

The limit of a function refers to the value that the function approaches, not the actual value (if any).

(1)Graphical

1

Not 0

Graphical Example

4

3

3.7 ish

Graphical Example

2

1.75

1

DNE

[0-3(1)]=-3

[2(2)]2=16

-1

1

= -2

One-sided limits approach from either the left or right side only.

left hand limit

right hand limit

value of the function

value of the two sided

limit

Graphical Example

2

1

1

2

3

4

At x=1:

does not exist

Graphical example

left hand limit

right hand limit

value of the function

2

1

1

2

3

4

At x=2:

1

GraphicalExample

DNE

4

2

3

2) Numerical Example

If I wanted to know f(1), you cant just plug it in. So use limits

Make a table of values from both sides

Around the value

as we can see, it approaches 3 from both sides, so…

Numerical Example

=0.5

Homework

p. 62 (1-6, 31,32,43,44,45,46)