Transcript Slide 1

Section 1-2
Finding Limits Graphically and
Numerically
A Group Project by:
Oliver “Jacko” Temple
Patrick “P Rawls” Rollings
Veejay “Jay” Costello
An Introduction to Limits
For every value other
than x=1, standard
curve-sketching
determines the y value.
How do we find the y
value at x=1?
x3  1
f (x) 
x 1
Numerical Approach
To fill in the missing value,
x=1, numerical
evaluation can be used.
x3  1
f (x) 
x 1
x
0.75
0.9
0.99
0.999
1
1.001
f(x) 2.311 2.710 2.970 2.997
?
3.003 3.030 3.310 3.813
f(x) approaches 3
1.01
1.1
1.25
f(x) approaches 3
Math Mumbo Jumbo [Important!]
Although x can’t equal 1 in that function, it
moves arbitrarily close to 1, and as a result
f(x) moves arbitrarily close to 3.
f (x)  3
Using Limit Notation it is written: lim
x 1
The limit of f(x) as x approaches 1 equals 3.
In general terms: lim f ( x )  L
x c
Example: Finding a Limit
1 for x  2
Graph f (x )  
0 for x  2
Believe it or not, the limit of f(x) as x approaches 2
is 1, even though it is defined as 0 at the point.
When Limits Fail You
From the left it seems
that the limit of f(x) at
x=0 is -1, and from the
right 1.
x
lim
x 0 x
Conflicting left-hand and
right-hand limits such
as this means that the
limit Does Not Exist
-1
1
Unbounded Behavior
Answer is not a finite
number, so the limit
does not exist.
1
lim 2
x 0 x
Oscillating Behavior
1
lim sin 
x 0
x
Limit does not exist because
the value switches
between -1 and 1.
x
2/π
sin(1/x)
1
2/(3 π) 2/(5 π) 2/(7 π) 2/(9 π) 2/(11 π) x0
-1
1
-1
1
-1
DNE
Hints that the limit DNE

the function approaches a different number from
the right than it does from the left

the function increases or decreases without bound
as x approaches c

the function jumps back and forth between two
fixed values as x approaches c
ε-δ Definition of a Limit
Let f be a function defined
on an open interval
containing p (except possibly
at p) and let L be a real
number. Consider:
lim f (x)  L
x p
This means that for any
choice of ε>0, there exists a
δ>0 such that
|x–p|<δ  |f(x)-L|<ε


ε is epsilon
δ is delta
Using delta
values you
can make a
range that
contains the y
value of x
approaching c
Use of Cauchy’s Definition
 Prove
theorems about limits
 Establish the existence or
nonexistence of particular limits
 Eventually we will learn an
easier way to find limits