Lesson03 The Concept Of Limit 027 Slides

Section 1.2–1.3
A Catalog of Essential Functions
The Limit of a Function

V63.0121.027, Calculus I

September 10, 2009

Announcements
Syllabus is on the common Blackboard
Ofﬁce Hours MTWR 3–4pm
Read Sections 1.1–1.3 of the textbook this week.
. . . . . .

Outline
Classes of Functions
Linear functions
Quadratic functions
Cubic functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
Limits
Heuristics
Errors and tolerances
Examples
Pathologies
. . . . . .

Classes of Functions

linear functions, deﬁned by slope an intercept, point and
point, or point and slope.
quadratic functions, cubic functions, power functions,
polynomials
rational functions
trigonometric functions
exponential/logarithmic functions

. . . . . .

Linear functions

Linear functions have a constant rate of growth and are of the
form
f(x) = mx + b.

. . . . . .

Linear functions

form
f(x) = mx + b.

Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5
mile. Write the fare f(x) as a function of distance x traveled.

. . . . . .

Linear functions

form
f(x) = mx + b.

Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5
mile. Write the fare f(x) as a function of distance x traveled.

Answer
If x is in miles and f(x) in dollars,

f(x) = 2.5 + 2x

. . . . . .

Quadratic functions

These take the form

f(x) = ax2 + bx + c

. . . . . .

Quadratic functions

These take the form

f(x) = ax2 + bx + c

The graph is a parabola which opens upward if a > 0, downward
if a < 0.

. . . . . .

Cubic functions

These take the form

f(x) = ax3 + bx2 + cx + d

. . . . . .

Other power functions

Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.

. . . . . .

Rational functions

Deﬁnition
A rational function is a quotient of polynomials.

Example
x 3 (x + 3 )
The function f(x) = is rational.
(x + 2)(x − 1)

. . . . . .

Trigonometric Functions

Sine and cosine
Tangent and cotangent
Secant and cosecant

. . . . . .

Exponential and Logarithmic functions

exponential functions (for example f(x) = 2x )
logarithmic functions are their inverses (for example
f(x) = log2 (x))

. . . . . .


Take the sine function and graph these transformations:
( π)
sin x +
( 2
π)
sin x −
2
π
sin (x) +
2
π
sin (x) −
2

. . . . . .


Take the sine function and graph these transformations:
( π)
sin x +
( 2
π)
sin x −
2
π
sin (x) +
2
π
sin (x) −
2
Observe that if the ﬁddling occurs within the function, a
transformation is applied on the x-axis. After the function, to the
y-axis.

. . . . . .

Vertical and Horizontal Shifts

Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units

y = f(x) − c, shift the graph of y = f(x) a distance c units

y = f(x − c), shift the graph of y = f(x) a distance c units

y = f(x + c), shift the graph of y = f(x) a distance c units

. . . . . .


upward



. . . . . .


upward
downward


. . . . . .


upward
downward
y = f(x − c), shift the graph of y = f(x) a distance c units to
the right

. . . . . .


upward
downward
y = f(x − c), shift the graph of y = f(x) a distance c units to
the right
y = f(x + c), shift the graph of y = f(x) a distance c units to
the left

. . . . . .

Composition is a compounding of functions in
succession

g
. ◦f
.
x
. f
. . g
. . g ◦ f)(x)
(
f
.(x)

. . . . . .

Composing

Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.

. . . . . .

Composing

Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.

Solution
f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the
same.

. . . . . .

Decomposing

Example
√
Express x2 − 4 as a composition of two functions. What is its
domain?

Solution √
We can write the expression as f ◦ g, where f(u) = u and
g(x) = x2 − 4. The range of g needs to be within the domain of f.
To insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.

. . . . . .

The Far Side

. . . . . .

Zeno’s Paradox

That which is in
locomotion must
arrive at the
half-way stage
before it arrives at
the goal.

(Aristotle Physics VI:9,
239b10)

. . . . . .

Heuristic Definition of a Limit

Definition
We write
lim f(x) = L
x→a

and say

“the limit of f(x), as x approaches a, equals L”

if we can make the values of f(x) arbitrarily close to L (as close to
L as we like) by taking x to be sufficiently close to a (on either
side of a) but not equal to a.

. . . . . .

The error-tolerance game

A game between two players to decide if a limit lim f(x) exists.
x→a
Player 1: Choose L to be the limit.
Player 2: Propose an “error” level around L.
Player 1: Choose a “tolerance” level around a so that
x-points within that tolerance level are taken to y-values
within the error level.
If Player 1 can always win, lim f(x) = L.
x→a

. . . . . .


L
.

.
a
.

. . . . . .


L
.

.
a
.

To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.

. . . . . .


T
. his tolerance is too big

L
.

.
a
.


. . . . . .


S
. till too big

L
.

.
a
.


. . . . . .


T
. his looks good

L
.

.
a
.


. . . . . .


S
. o does this

L
.

.
a
.


. . . . . .


L
.

.
a
.

If Player 2 shrinks the error, Player 1 can still win.
. . . . . .

Example
Find lim x2 if it exists.
x→0

. . . . . .

Example
x→0

Solution
I claim the limit is zero.

. . . . . .

Example
x→0

Solution
If the error level is 0.01, I need to guarantee that
−0.01 < x2 < 0.01 for all x sufﬁciently close to zero.

. . . . . .

Example
x→0

Solution
If −0.1 < x < 0.1, then 0 ≤ x2 ≤ 0.01, so I win that round.

. . . . . .

Example
x→0

Solution
What should the tolerance be if the error is 0.0001?

. . . . . .

Example
x→0

Solution
What should the tolerance be if the error is 0.0001?
By setting tolerance equal to the square root of the error, we can
guarantee to be within any error.

. . . . . .

Example
|x|
Find lim if it exists.
x→0 x

. . . . . .

Example
|x|
x→0 x

Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0

What would be the limit?

. . . . . .

y
.

. .
1

. x
.

. 1.
−

. . . . . .

y
.

. .
1

. x
.

.
Part of graph in-
. 1.
− side blue is not
inside green

. . . . . .

y
.

.
Part of graph in-
side blue is not . .
1
inside green

. x
.

. 1.
−

. . . . . .

y
.

.
Part of graph in-
side blue is not . .
1
inside green

. x
.

. 1.
−

These are the only good choices; the limit does not exist.

. . . . . .

One-sided limits

Deﬁnition
We write
lim f(x) = L
x→a+

and say

“the limit of f(x), as x approaches a from the right, equals L”

side of a) and greater than a.

. . . . . .

One-sided limits

Deﬁnition
We write
lim f(x) = L
x→a−

and say

“the limit of f(x), as x approaches a from the left, equals L”

side of a) and less than a.

. . . . . .

Example
|x|
x→0 x

Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0

What would be the limit?
The error-tolerance game fails, but

lim f(x) = 1 lim f(x) = −1
x→0+ x→0−

. . . . . .

Example
1
x→0+ x

. . . . . .

y
.

.? .
L

. x
.
0
.

. . . . . .

y
.

.
The graph escapes the
green, so no good

.? .
L

. x
.
0
.

. . . . . .

y
.

E
. ven worse!

.? .
L

. x
.
0
.

. . . . . .

y
.

.
The limit does not exist
because the function is
unbounded near 0

.? .
L

. x
.
0
.

. . . . . .

Example
1
x→0+ x
Solution
The limit does not exist because the function is unbounded near
0. Next week we will understand the statement that
1
lim = +∞
x→0+ x

. . . . . .

Weird, wild stuff

Example (π )
Find lim sin if it exists.
x→0 x

. . . . . .

Weird, wild stuff

Example (π )
x→0 x

f(x) = 0 when x =

f(x) = 1 when x =

f(x) = −1 when x =

. . . . . .

Weird, wild stuff

Example (π )
x→0 x
1
f(x) = 0 when x = for any integer k
k
f(x) = 1 when x =

f(x) = −1 when x =

. . . . . .

Weird, wild stuff

Example (π )
x→0 x
1
k
1
2k + 1/2
f(x) = −1 when x =

. . . . . .

Weird, wild stuff

Example (π )
x→0 x
1
k
1
2k + 1/2
1
f(x) = −1 when x = for any integer k
2k − 1/2

. . . . . .

Weird, wild stuff continued

Here is a graph of the function:
y
.
. .
1

. x
.

. 1.
−

There are inﬁnitely many points arbitrarily close to zero where
f(x) is 0, or 1, or −1. So the limit cannot exist.

. . . . . .

What could go wrong?
Summary of Limit Pathologies

How could a function fail to have a limit? Some possibilities:
left- and right- hand limits exist but are not equal
The function is unbounded near a
Oscillation with increasingly high frequency near a

. . . . . .

Meet the Mathematician: Augustin Louis Cauchy

French, 1789–1857
Royalist and Catholic
made contributions in
geometry, calculus,
complex analysis,
number theory
created the deﬁnition of
limit we use today but
didn’t understand it

. . . . . .

Precise Deﬁnition of a Limit
No, this is not going to be on the test

Let f be a function deﬁned on an some open interval that
contains the number a, except possibly at a itself. Then we say
that the limit of f(x) as x approaches a is L, and we write

lim f(x) = L,
x→a

if for every ε > 0 there is a corresponding δ > 0 such that

if 0 < |x − a| < δ , then |f(x) − L| < ε.

. . . . . .

The error-tolerance game = ε, δ

L
.

.
a
.

. . . . . .


L
. +ε
L
.
. −ε
L

.
a
.

. . . . . .


L
. +ε
L
.
. −ε
L

.
. −δ
a a
. a
. +δ

. . . . . .


T
. his δ is too big
L
. +ε
L
.
. −ε
L

.
. −δ
a a
. a
. +δ

. . . . . .


L
. +ε
L
.
. −ε
L

.
a a
. −δ . . +δ
a

. . . . . .


T
. his δ looks good
L
. +ε
L
.
. −ε
L

.
a a
. −δ . . +δ
a

. . . . . .


S
. o does this δ
L
. +ε
L
.
. −ε
L

.
aa
. −δ . . +δ
a

. . . . . .

Lesson03 The Concept Of Limit 027 Slides

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