Class notes calculus unit 1

unit 1
limits and rates of change




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, Experiencing Limits
Evaluating Limits of Convergent Sequences

A sequence is a function whose domain is the set of positive integers n=1,2,3,….

The values or individual terms of a sequence are generally denoted by a subscript
of n on t. In other words, we use tn rather than f(n).
For example, the list of all positive odd numbers forms the sequence 1,3,5,7,….
This sequence could be represented algebraically by two different formulas:
1. Recursive Formula
tn  tn1  2 where t1  1
2. Arithmetic Formula
t n  1  2 n  1 
 2n  1 for n  1,2,3,...
If we continue this sequence of numbers, would this sequence approach a single
value?
In other words, as n→+∞ does tn approach a limit?
As n increases, we see that tn becomes arbitrarily large in value.
Therefore, as n→+∞, tn→+∞.
We could use limits to write this as
lim (2n  1)  
n 

The behavior of infinite sequences

It is often very important to examine what happens to a sequence as n gets very
large. There are three types of behavior that we shall wish to describe explicitly.
These are

 sequences that ‘tend to infinity’;
 sequences that ‘converge to a real limit’;
 sequences that ‘do not tend to a limit at all’.

First we look at sequences that tend to infinity. We say a sequence tends to infinity
if, however large a number we choose, the sequence becomes greater than that
number, and stays greater. Here are some examples of sequences that tend to
infinity.


an  n2 ;n  1 n an  n2
1
2
5
10
15


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,Now we look at sequences with real limits. We say a sequence tends to a real limit if
there is a real number, L, such that the sequence gets closer and closer to it. We say
L is the limit of the sequence.

1
an  ; n  1
n n 1
an =
n
1
2
5
10
50




And finally we look at sequences that cannot approach any specific number L as n
grows large.
n  nπ 
a n = sin  
 6 
0
3
6
9
12




Definitions:
We say that the sequence {an} converges (or is convergent or has limit)
if it tends to a number L.
A sequence diverges (or is divergent) if it does not tend to any number.




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, 1. 1 The limit of a Function

Calculus has been called the study of continues change, and the limit is
the basic concept that describe and analyze such change.
The limit of a function describes the behaviour of the function when The
variable is near, but does not equal, a specific number ( Fig.1)
If the values of f(x) get closer and closer, as close as we want, to one
number L as we take values of x very close to ( but not equal to) a
number c ,then we say:


()
The limit of f(x), as x approaches c is L and we write: lim f x = L .
x®c


f(c) is the ONLY number that describes the behaviour (value) of f(x) AT the point x=c.
lim f  x  is a single number that describes the behaviour of f NEAR, BUT NOT AT point
x c

x=c.

Example #1:
Use the graph of y=f(x) and determine the
following limits.
(a) lim f  x  
x2



(b) lim f  x 
x 3


(c) lim f  x  
x 1



(d) lim f  x  
x4




Example #2: Graph each function, then complete the Table of values to find each limit.

1
(a) lim
x 2 x

x 1.9 1.99 1.999 2 2.0001 2.001 2.01

f(x)




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