Transition Maths and Algebra with Geometrytomtracz/TRANSITION/09 functions and limits.pdf · Basic De nitions Elementary functions Limits Asymptotes Transition Maths and Algebra with

Basic DefinitionsElementary functions

LimitsAsymptotes

Transition Maths and Algebra with Geometry

Tomasz Brengos

Lecture NotesElectrical and Computer Engineering

Tomasz Brengos Transition Maths and Algebra with Geometry 1/36


LimitsAsymptotes

Contents

1 Basic Definitions

2 Elementary functions

3 Limits

4 Asymptotes



LimitsAsymptotes

Injective functions

Recall the following definition:

Definition

A function f : X → Y is injective (1-1) if for any two argumentsx1, x2 ∈ X we have

f (x1) = f (x2) =⇒ x1 = x2.

Examples:sin : R→ R is not 1-1, since sin(0) = sin(π).sin : [−π

2 ,π2 ]→ R is 1-1.



LimitsAsymptotes

Inverse functions

Recall the following:

Definition

If a function f : A→ B is 1-1 then we define a function f −1 : f (A)→ A, whosedomain is the range of f and whose codomain is the domain of f , by:

f −1(b) = a if f (a) = b.

f −1 is called the inverse of f .

Fact

If the function f : A→ B is 1-1 then for any a ∈ A the function f and itsinverse f −1 satisfy:

(f −1 ◦ f )(a) = f −1(f (a)) = a

Moreover, for any b ∈ f (A) we have

(f ◦ f −1)(b) = f (f −1(b)) = b



LimitsAsymptotes

Contents

1 Basic Definitions


3 Limits

4 Asymptotes



LimitsAsymptotes

Polynomials and rational functions

Definition

Let n be a natural number and let a0, a1 . . . an ∈ R be contant coefficients. Afunction of the form

f (x) = an · xn + . . . + a1x + a0

is a polynomial function. Domain of any polynomial is the set of real numbers

R.

Definition

A function f (x) is called a rational function if it is of the form

f (x) =p(x)

q(x),

where p(x) and q(x) are polynomials. Domain depends on zeros of q(x).



LimitsAsymptotes

Exponential functions

Definition

Let a > 0 be a real number. A function defined by

f (x) = ax

is called an exponential function with base a. Its domain is the setof all real numbers.

Graph

Image source: wolframalpha.comTomasz Brengos Transition Maths and Algebra with Geometry 7/36


LimitsAsymptotes


The most important exponential function is fora = e = 2.7182818 . . .. Recall that

ex = limn→∞

(1 +

x

n

)n



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Fact

For any x , y ∈ R we have

ax · ay = ax+y .

Fact

For any a > 0 the function

f (x) = ax

is injective. Its range is given by (0,∞). Hence, the function isinvertible.



LimitsAsymptotes

Logarithmic functionsDefinition

Let a > 0 be any real number. A function loga, whose domain is(0,∞), which is defined by

loga x = y ⇐⇒ ay = x ,

is called a logarithmic function. Coefficient a is called a base.

Fact

The function f (x) = loga(x) is the inverse of the exponential function

g(y) = ay .

Fact

For any x ∈ (0,∞) and for any y ∈ R we have

aloga(x) = x loga(ay ) = y .



LimitsAsymptotes

Logarithmic functions

Notation

if a = e then loge(x) is denoted by ln(x),

if a = 10 then log10(x) is denoted by log(x),

if a = 2 then log2(x) is denoted by lg(x).

Fact

For any x ∈ (0,∞) and for any y ∈ R we have

e ln(x) = x ln(ey ) = y .



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Graph

Image source: wolframalpha.com



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Example:

log2 8 = 3, log31

9= −2

Properties

Let x , y be two positive real numbers.

aloga b = b,

loga 1 = 0,

loga a = 1,

loga(x · y) = loga(x) + loga(y),

loga( xy ) = loga(x)− loga(y),



LimitsAsymptotes


Properties

loga(xc) = c · loga(x),

loga b = 1logb a

for b > 0,

logan b = 1n loga b,

loga b = logc blogc a

for b, c > 0.



LimitsAsymptotes

Trigonometric functions

Sine

y = sin(x). Domain of sin(x) is the set of all real numbers R.




LimitsAsymptotes


Cosine

y = cos(x). Domain of cos(x) is the set of all real numbers R.




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Tangent

y = tan(x). Domain of tan(x) is the following set D:

D = R \ {(2k + 1)π

2| k ∈ Z}




LimitsAsymptotes


Cotangent

y = cot(x). Domain of cot(x) is the following set D:

D = R \ {2k · π | k ∈ Z}




LimitsAsymptotes

Cyclometric functions

Arcsine

y = arcsin(x). Domain of arcsin(x) is the set [−1, 1]. It is definedas the iverse of sin restricted to the interval [−π

2 ,π2 ]. Range of

arcsin is [−π2 ,

π2 ].




LimitsAsymptotes


Arccosine

y = arccos(x). Domain of arccos(x)is the set [−1, 1]. It is definedas the iverse of cos restricted to the interval [0, π]. Range of arccosis [0, π].




LimitsAsymptotes


Arctangent

y = arctan(x). Domain of arctan(x) is the set R. It is defined asthe iverse of tan restricted to the interval (−π

2 ,π2 ). Range of

arctan is (−π2 ,

π2 ).




LimitsAsymptotes

Hyperbolic functions

Hyperbolic functions

sinh x =ex − e−x

2

cosh x =ex +−e−x

2

tanh x =sinh x

cosh x=

ex − e−x

ex + e−x

coth x =1

tanh x=

ex + e−x

ex − e−x



LimitsAsymptotes

Contents

1 Basic Definitions


3 Limits

4 Asymptotes



LimitsAsymptotes

Limit definition

Definition

Number L is a limit of a function y = f (x) at point a if for anysequence {xn} of elements different from a belonging to thedomain of f and convergent to a the sequence of values {f (xn)}converges to L. In other words, L ∈ R is a limit at a if

xnn→∞→ a and xn 6= a =⇒ f (xn)

n→∞→ L

L is a common limit of all sequences of values {f (xn)} for xnconvergent to a and different from a. If such a common valuedoesn’t exists then the limit doesn’t exist.



LimitsAsymptotes

Notation

Notation

If L is a limit of y = f (x) at a then we denote it by

limx→a

f (x) = L

Example: Consider f (x) = x2 and let a = 2. Pick any sequence{xn} converging to 2 and different from 2. We have

(xn)2 = xn · xnn→∞→ 2 · 2 = 4.

Hence,limx→2

x2 = 4



LimitsAsymptotes

Example

Consider a function

f (x) =

−1 for x < 00 for x = 01 for x > 0

If we pick a sequence xn > 0 and convergent to 0 we have

f (xn) = 1→ 1.

Now for a sequence x ′n < 0 and convergent to 0 we have

f (x ′n) = −1→ −1.

Hence, the limit at 0 of f doesn’t exist.



LimitsAsymptotes

One-sided limits

Definition

Number L is a right-sided (left-sided) limit of a function y = f (x)at point a if for any sequence {xn} convergent to a with xn > a(resp. xn < a) the sequence of values {f (xn)} converges to L.One-sided limits are denoted by

limx→a+

f (x) = L limx→a−

f (x) = L.



LimitsAsymptotes

Example

Consider a function

f (x) =

−1 for x < 00 for x = 01 for x > 0

We see thatlim

x→0+f (x) = 1 lim

x→0−f (x) = −1



LimitsAsymptotes

One-sided limits

Fact

If either limx→a+ f (x) or limx→a− f (x) doesn’t exist thenlimx→a f (x) doesn’t exist either.

Theorem

limx→a f (x) exists if and only if limx→a+ f (x) and limx→a− f (x)exist and

limx→a+

f (x) = limx→a−

f (x)



LimitsAsymptotes

Limits at ∞

Definition

By limx→∞ f (x) we denote the common limit (if it exists) of thesequences {f (xn)} for which limn→∞ xn =∞. Similarily, we definelimx→−∞ f (x)



LimitsAsymptotes

Properties of limits

Properties

Let y = f (x) and y = g(x) be two functions and assume thatlimx→a f (x) and limx→a g(x) exist. Then:

limx→a(f (x) + g(x)) = limx→a f (x) + limx→a g(x),

limx→a(f (x)− g(x)) = limx→a f (x)− limx→a g(x),

limx→a(f (x) · g(x)) = limx→a f (x) · limx→a g(x),

limx→af (x)g(x) = limx→a f (x)

limx→a g(x) for g(x) 6= 0 around a and

limx→a g(x) 6= 0.

Remark

The above theorem is true if we replace a with ∞ or −∞ and limits with

one-sided limits



LimitsAsymptotes

Sandwich Theorem for functions

Sandwich Theorem

Let three functions y = f (x), y = g(x) and y = h(x) satisfy

f (x) ≤ g(x) ≤ h(x)

is some neighbourhood of a and letlimx→a f (x) = limx→a h(x) =: L. Then limx→a g(x) exists and

limx→a

g(x) = L.

Remark

The above theorem is true if we replace a with ∞ or −∞ and limits with

one-sided limits



LimitsAsymptotes

Contents

1 Basic Definitions


3 Limits

4 Asymptotes



LimitsAsymptotes

Asymptotes

Definition

A line y = b is a horizontal asymptote of the function y = f (x) ifeither

limx→∞

f (x) = b or limx→−∞

f (x) = b

Definition

A line x = a is a vertical asymptote of the function y = f (x) ifeither

limx→a+

f (x) = ± or limx→a−

f (x) = ±



LimitsAsymptotes

Asymptotes

Definition

A line y = ax + b is an oblique asymptote of the function y = f (x)if either

limx→∞

[f (x)− (ax + b)] = 0 or limx→−∞

[f (x)− (ax + b)] = 0

Remark

Any horizontal asymptote is also an oblique asymptote for a = 0.



LimitsAsymptotes

Asymptotes

Theorem

A line y = ax + b is an oblique asymptote of y = f (x) at ∞ if andonly if

a = limx→∞

f (x)

x,

b = limx→∞

[f (x)− a · x ]

Example: consider y = 1x , y = ex and y = x2−1

x .


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