transition maths and algebra with geometrytomtracz/transition/09 functions and limits.pdf · basic...

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 .



LimitsAsymptotes


Graph

Image source: wolframalpha.com



LimitsAsymptotes


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.




LimitsAsymptotes


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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