MATH 2144 Simmons
2.7: Limits at Infinity
So far weve been considering limits as , where is a finite number, but what
if we wanted to determine the long term behavior of a function?
Notation:
;
increases without bound.
;
decreases without bound.
Examp

MATH 2144 Simmons
2.6: Trigonometric Limits
Important Trigonometric Limits:
sin
=
0
1 cos
=
0
lim
lim
Recall from 2.2: lim 0
Example 1: Evaluate lim 0
sin
3
sin
=
.
2.6 - Page 1 of 6
MATH 2144 Simmons
Example 2: Evaluate lim 0
sin 7
7
.
WARNING! The t

MATH 2144 Simmons
2.8: Intermediate Value Theorem
Consider the following two cases:
A hiker spends 3 days hiking up
Mount Kilimanjaro, which has an
elevation of 5895 meters.
Can we guarantee that at some point in his 3 day journey, the hiker was at an
ele

MATH 2144 Simmons
2.4: Limits and Continuity
Continuous: continuing without stopping: happening or existing
without a break or interruption.
The following are graphs of continuous functions;
while these are examples of discontinuous functions.
Question: W

MATH 2144 Simmons
Precalculus Review: Inverse Functions
Section 1.5
Informally the inverse of (), denoted 1 (), is the function that reverses the
effect of ().
Look at inputs and outputs of two related functions:
g ( x)
f ( x) 3 4 x
x 3
4
Input x
f(x)
Ou

MATH 2144 Simmons
2.2: Limits: A Numerical and Graphical Approach
When the values successively attributed to the same variable
approach a fixed value indefinitely, in such a way as to end up
differing from it by as little as one could wish, this last valu

MATH 2144 Simmons
2.5: Evaluating Limits Algebraically
Recall: In Example 4 from section 2.2 we discovered lim 1
2 4+3
1
= 2 by
making a chart of values. Compare the following two graphs illustrating an
important feature of this type of function:
() =
2

MATH 2144 Simmons
2.1: Limits, Rates of Change, and Tangent Lines
Calculus is a large subject usually divided into two branches, differential and
integral calculus. Perhaps the most basic concept of differential calculus is using
the limit concept to find

MATH 2144 Simmons
2.3: Basic Limit Laws
Sum Law:
If lim () and lim () exist, then;
lim() + () = lim () + lim ().
xc
Example 1: Let () = 3 , and () = . Use the sum law to evaluate
lim () + ().
x1
Constant Multiple Law:
If lim () and lim () exist, then for

MATH 2144 Simmons
Precalculus Review: The Basics
Section 1.1
Real Numbers: numbers represented by a decimal expansion.
3
= 0.375
8
Finite
Repeating
Nonrepeating
1
= 0.142857142857 = 0. 142857
7
= 3.141592653589793
Number Lines and Interval Notation
Set