1. Teaching Limits so that
Students will Understand Limits
Presented by
Lin McMullin
National Math and Science Initiative
2.
3 2
4 6
2
x x x
f x
x
Continuity
What happens at x = 2?
What is f(2)?
What happens near x = 2?
f(x) is near 3
What happens as x approaches 2?
f(x) approaches 3
3. What happens at x = 1?
What happens near x = 1?
As x approaches 1, g increases without bound, or g approaches infinity.
As x increases without bound, g approaches 0.
As x approaches infinity g approaches 0.
Asymptotes
2
1
1
g x
x
6.
The Area Problem
2
1
0
1
3
h x x
j x
x
x
What is the area of the outlined region?
As the number of rectangles increases with out bound, the area of the
rectangles approaches the area of the region.
7. The Tangent Line Problem
What is the slope of the black line?
As the red point approaches the black point, the red secant line approaches
the black tangent line, and
The slope of the secant line approaches the slope of the tangent line.
8. As x approaches 1, (5 – 2x) approaches ?
f(x)
within
0.08
units
of
3
x
within
0.04
units
of
1
f(x)
within
0.16
units
of
3
x
within
0.08
units
of
1
0.90 3.20
0.91 3.18
0.92 3.16
0.93 3.14
0.94 3.12
0.95 3.10
0.96 3.08
0.97 3.06
0.98 3.04
0.99 3.02
1.00 3.00
1.01 2.98
1.02 2.96
1.03 2.94
1.04 2.92
1.05 2.90
1.06 2.88
1.07 2.86
1.08 2.84
1.09 2.82
x 5 2x
9.
1
lim 5 2 3
x
x
1
2
2 2
2 2
5 2 3
5 2 3
x
x
x
x
x
f x L
10.
1 or 1 1
2 2 2
x x
5 2 3
or
3 5 2 3
x
x
1
lim 5 2 3
x
x
Graph
12. When the values successively attributed
to a variable approach indefinitely to a
fixed value, in a manner so as to end by
differing from it as little as one wishes,
this last is called the limit of all the
others.
Augustin-Louis Cauchy (1789 – 1857)
The Definition of Limit at a Point
13.
lim if, and only if, for any number 0
there is a number 0 such
if 0 ,
t
then
hat
x a
x
L
f x L
f
a
x
The Definition of Limit at a Point
Karl Weierstrass (1815 – 1897)
14.
lim if, and only if, for any number 0
there is a number 0 such
if 0 ,
t
then
hat
x a
x
L
f x L
f
a
x
lim if, and only if, for any number 0
there is a numb
i
er 0 and such t
f , then
hat
x a
a x a L f x L
f x L
x a
The Definition of Limit at a Point
Karl Weierstrass (1815 – 1897)
15.
lim 0 0 such that
, whenever 0
x a
f x L
f x L x a
Footnote:
The Definition of Limit at a Point
16. Footnote:
The Definition of Limit at a Point
lim 0 0 such
whe
that
, 0
never
x a
f x L
f x L x a
21.
2
3
lim 9
x
x
2
9
3 3
x
x x
Near 3, specifically in (2,4), 5 3 7
x x
7 3
3
7
x
x
22.
2
3
lim 9
x
x
2
9
3 3
x
x x
Near 3, specifically in (2,4), 5 3 7
x x
7 3
3
7
x
x
the smaller of 1 and
7
Graph
23.
lim sin( ) sin( )
x a
x a
Graph
a + delta
a - delta
Length of Red Segment = | sin(x) - sin(a
Length of Blue Arc = | x - a |
Therefore, | sin(x) - sin(a) | < | x - a |
sin x
sin a
(cos x, sin x)
(cos a, sin a)
(1, 0)
x in radians
(0,1)
24.
lim if, and only if, for any number 0
there is a number 0 such that
if 0 , then
lim if, and only if, for any number 0
there is a number 0 such that
if 0 , t
a
a
x
x
f x L
f x
x a L
f x L
a x
hen f x L
One-sided Limits
25. Limits Equal to Infinity
lim if, and only if, for any number
there is a number such that
if
0
Graphically this is a vertical asymptote
0 , then
x a
M
f x M M
f x
x a f x
lim if, and only if, for any number
there is a number such that
if 0 , then
0
x a
f x
x
M
f x
a M
26. Limit as x Approaches Infinity
lim if, and only if, for any number 0
there is a number suc
Graphically, this is a horizontal a
h that
if , then
sym t
0
p ote
x
f x L
M f x L
M
x
lim if, and only if, for any number 0
there is a number such that
if , then
0
x
M
x
f x L
f x L
M
27. Limit Theorems
Almost all limit are actually found by substituting the
values into the expression, simplifying, and coming up
with a number, the limit.
The theorems on limits of sums, products, powers, etc.
justify the substituting.
Those that don’t simplify can often be found with more
advanced theorems such as L'Hôpital's Rule
28. The Area Problem
2
1
0
1
3
h x x
j x
x
x
29. The Area Problem
2
2 2 2 2
2
2 2
1
2
2 2
1
length 1
3 1 2
width
x-coordinates = 1,1 ,1 2 ,1 3 , ...,1
Area 1 1
32
Area =lim 1 1
3
n n n n
n
n n
i
n
n n
n
i
h x j x x
b a
n n n
n
i
i
30. The Area Problem
2
2 3
2 3
2 2
2 2 2 2 4 2 4
1 1 1 1
8 8
4 2
1
1 2 1
6
1
8 8
4
8
3
32
1
3
2
1
lim 1 1 lim 2 lim lim
lim lim lim
lim lim lim
4 4
1
n n n n
n n n n n n n
i n n n
i i i i
n n n
n n n
n n n
n
i
n
n n n
n
n n
i
n
n
n
i
i
i i
n
i
i
31.
,
P a x f a x
,
T a f a
x
f a x f a
y f a x a
m
The Tangent Line Problem
As
0
slope
P T
x
PT
f a x f a
a x a
m
m
32.
,
P a x f a x
,
T a f a
x
f a x f a
a
y f a x a
f
The Tangent Line Problem
As
0
slope m
P T
x
PT
f
f
a x f a
a x
a
a
33.
34. Lin McMullin
National Math and Science Initiative
325 North St. Paul St.
Dallas, Texas 75201
214 665 2500
lmcmullin@NationalMathAndScience.org
www.LinMcMullin.net Click: AP Calculus
36.
Let
sin sin
sin sin
limsin sin
x a
x a x a
x a x a
x a
37.
x a
f x L
1
lim 5 2 3
x
x
lim if, and only if,
for any number 0 there is a
number 0 such that if
0 , then
x a
f x L
x a f x L
38.
,
P a x f a x
,
T a f a
x
f a x f a
y f a x a
m
The Tangent Line Problem
As
0
slope
P T
x
PT
f a x f a
a x a
m
m
