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Math 1314
Day 1 Notes
Instructor:
Marjorie Marks
Email:
[email protected]
.
CASA Website:
casa.uh.edu
My personal website:
math.uh.edu/~mmarksc
Prerequisites:
You must have credit for College Algebra.
You can earn credit by placing
out of Math 1310 on the UH Math Placement Test or by successfully completing College
Algebra here or at another institution.
Not only do you need to have credit in that class, I expect you to remember just about
everything from College Algebra.
There are lots of online resources to help you, but if you don’t remember much of
College Algebra, this class may not be for you.
Go to my personal website,
www.math.uh.edu/~mmarksc
.
Click on the link to Math
1314.
There’s a link to “Review Materials.”
On the resulting page, there’s a link to
Prerequisite Review.
There are worked problems and flash videos of review problems.
You need to know virtually everything on that review.
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View Full Document Section 10.4  Limits
What is calculus?
The body of mathematics that we call calculus resulted from the investigation of two
basic questions by mathematicians in the 18
th
century.
1.
How can we find the line tangent to a curve at a given point on the curve?
2.
How can we find the area of a region bounded by an arbitrary curve?
The investigation of each of these questions relies on the process of finding a
limit
, so
we’ll start by informally defining a limit and follow that by learning techniques for
finding limits.
Limits
Informal definition:
Finding a limit amounts to answering the following question:
What is happening to the
y
value of a function as the
x
value approaches a specific target
number?
If the
y
value is approaching a specific number, then we can state the limit of
the function as
x
gets close to the target number.
Look at these graphs and find the limit as x gets really close to 1 in both cases.
4
3
2
1
1
2
3
4
4
3
2
1
1
2
3
4
x
y
4
3
2
1
1
2
3
4
4
3
2
1
1
2
3
4
x
y
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View Full Document Example 1:
Find
2
lim ( )
x
f x
→
.
Find
0
lim ( )
x
f x
→
.
4
3
2
1
1
2
3
4
4
3
2
1
1
2
3
4
x
y
It does not matter whether or not the
x
value every reaches the target number.
It might, or
it might not!
Example 2
:
Find
1
lim ( )
x
f x
→
.
4
3
2
1
1
2
3
4
5
4
3
2
1
1
2
3
4
x
y
When can a limit fail to exist?
We will look at two cases where a limit fails to exist (note:
there are more, but some are
beyond the scope of this course).
Case 1
:
The
y
value approaches one number from numbers smaller than the target
number and it approaches a second number from numbers larger than the target number:
Case 2
:
At the target number for the
x
value, the graph of the function has a vertical
asymptote.
For either of these two cases, we would say that the limit as
x
approaches the target
number “does not exist.”
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View Full Document More Formal Definition
:
We say that a function
f
has
limit
L
as
x
approaches the target number
a
, written
L
x
f
a
x
=
→
)
(
lim
if the value
f
(
x
) can be made as close to the number
L
as we please by taking
x
sufficiently close to (but not equal to)
a
.
Note that
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This note was uploaded on 02/21/2012 for the course MATH 1314 taught by Professor Marks during the Fall '08 term at University of Houston.
 Fall '08
 MARKS
 Algebra

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