A.D.
M R Richards
1
Functions And Limits
Basics
Definitions:
If two variables x and y follow a rule, “when x is given then y is determined as …”
Then y is said to be a function of x, this can be written as y=f(x):
Where x is the independent variable
and y is the dependant variable.
e.g. The area of a circle is a function of the radius as A(r )  r 2
Range And Domain
For a given set of x (the domain) there is a corresponding set of y values (the range).
e.g. A(r )  r 2
Domain:
Range:
0<r<2
0 < A < 4
Notation
y=f(x) is the same as y=y(x)
Special Functions
Polynomials
A polynomial of the n degree is given by:
y  a0  a1 x  a 2 x 2  ...a n x n
n
y   an x n
n o
Linear Functions (1st Degree Polynomial)
A linear function is of the form:
y
y = a0 + a1x
x
The y-intercept is given by a0 and the gradient (slope) is given by a1
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A.D.
M R Richards
Quadratic Functions (2nd Degree Polynomial)
A quadratic function is either of the form:
y = a0 + a1x + a2x2
a2 > 0
y  ax
a2 < 0

Assuming x  0,   1
Top, middle, bottom:
0   1
 1
 1
Assuming x  0,   0
Heaviside Step Function
0, x  0
H ( x)  
Whenever (x) is negative H(x)=0
1, x  0
H(x-1)
The derivative of H(x) is  (x) the Dirac Delta Function and is 0 everywhere apart
from one point where it is defined as tending towards infinity.
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A.D.
M R Richards
The main use of this function is modelling on/off forces such as voltages: i.e.
v(t )  H (t ) sin t
Modulus Function
 x, x  0
x 
 x, x  0
Even And Odd Functions
f(x) is even if f(x)=f(-x), i.e. it has symmetry about the y axis.
e.g. y=x2, x4, cos x
f(x) is odd if f(x)=-f(-x), i.e. it has pinhole symmetry, (The reflected function in y then
x gives the same function).
e.g. y=x, x3, sin x
An even function multiplied by an even function gives an even function.
An odd function multiplied by an odd function gives an even function.
An even function multiplied by an odd function gives an odd function.
Functions need not be even or odd.
a

For even functions
a
a
For odd functions
 f ( x)dx  0
a
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a
f ( x)dx  2 f ( x)dx
0
A.D.
M R Richards
Inverse Functions
A function y=f(x) can be inverted to get x in terms of y, i.e. x=g(y), hence g is the
inverse of f.
e.g. Find the inverse of f(x)=x2
So
where x >0.
y=x2
x y
x y
Thus the inverse g ( y)  y .
It does not matter what variable we use, the inverse could equally be stated as
g ( x)  x . N.B that the inverse function is the function reflected in the line y=x.
y x
yx
i.e.
y  x2
Notation
The inverse function is often written as f-1(x)
Hence is f(x)=x2 f 1 ( x)  x
Question:
What is f(f-1(x))
In above case f(x)=x2
Thus f(f-1(x)) = x
=x
2
This is a general result  (for all) functions.
f(f-1(x) = x
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A.D.
M R Richards
Function Of A Function
Given two functions: f(x) and g(x) one can calculate a function of a function.
e.g. If f(x)=x2 and g(x)=sin x.
Then f(g(x), (putting g into f) gives:
f ( g ( x))  f (sin x)  sin 2 x
Note that this is different to g(f(x)):
g ( f ( x))  sin( x 2 )
Many Valued Functions
Consider y=sin x and its inverse y=sin-1x or y=arcsine x
For y=sin x there is only 1 value of y for each x.
However, for sin-1x for each value of x there are an infinite number of values for y.
Hence we say that the principle value of sin-1 x lies in the domain of


  sin 1 x  , the same logic can be followed for y=cos-1x (y=arcos x).
2
2
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A.D.
M R Richards
Logarithms, Exponentials And Hyperbolic Functions
The natural logarithm, ln x or logex is defined as:
x
dt
ln x   , x  0
t
1
From this it follows that
d
1
ln x 
dx
x
Hence:
ln 1  0
ln x1 x2  ln x1  ln x2
Proof that ln x1 
x1

1
dr
t
Introducing the variable s gives:
s  tx2
ds  dtx2
ds dt

s
t
Hence:
ln x1 
x1 x2

x2
ds

s
x1 x2

1
x
ds 2 ds

s 1 s
ln x1  ln( x1 x2 )  ln x2QED
It follows that:
ln x 1   ln x
ln x n  n ln x
As x tends towards 0, y tends towards negative infinity.
As x tends towards infinity, y tends towards infinity.
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A.D.
M R Richards
Exponential Function
Consider x=ln y
What is the function such that y=f(x), i.e. the inverse of ln x.
Write x1=ln y1
y1=f(x1)
x2=ln y2
y2=f(x2)
Hence x1 + x2 = ln y1 + ln y2
= ln y1y2
Therefore y1y2=f(x1+x2)
Therefore f must satisfy:
f(x1)f(x2)=f(x1+x2)
This is only valid if f(x)=ax, where a is just some number.
To determine a, we must look at derivatives.
dx 1
dy
 
y
dy y
dx
So what value of a gives:
d x
a  ax
dx
The value is 2.71828183…
This is symbolised by e and is one of Euler’s numbers.
Hence the inverse of ln x is y=ex
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A.D.
M R Richards
Hyperbolic Functions
e x  ex
cosh x 
2
Even function
sinh x 
e x  ex
2
Odd function
tanh x 
sinh x
cosh x
1
e x  e x
ex  ex
-1 Odd function
Relationships Between Functions
1
coth x 
tanh x
1
cosh 2 x  sinh 2 x  1
sec hx 
cosh x
cosh 2 x  sinh 2 x  cosh 2 x
1
cos ech 
sinh x
Differentiation Of Hyperbolic Functions
d
sinh x  cosh x
dx
d
cosh x  sinh x
dx
d
tanh x  sec h 2 x
dx
These similarities to trigonometric functions are not a coincidence as it can be shown
that cosh ix = cos x
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A.D.
Inverse Hyperbolic Function
If
y = sinh-1 x it means
x =sinh y
-1
y = cosh x
it means
x = cosh y
y  sinh 1 x  sinh y  x
sinh y  x
1 y
(e  e  y )  x
2
e y  2 xe y  1  0
2
e y  x 1  x 2
y  ln( x  1  x 2 )
i.e.
sinh 1 x  ln( x  1  x 2 )
cosh 1 x  ln( x  x 2  1
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M R Richards
A.D.
M R Richards
Limits And L’Hopitals Theorem
sin x
Consider f ( x) 
this function is defined everywhere except at x=0, where the
x
function becomes 0/0
However if you plot the function f(x) one can see it gets closer to 1 as x gets closer to
0.
Geometric Proof that
sin x
 1 as x  0
x
Consider a circle of radius 1:
A D
tan x
x
O
cos x
sin x
B
Area Of Triangle ODB > Area Of Sector OAB > Area Of Triangle OAB
½tan x
>
½x
>
½sin x
Dividing by ½sin x gives:
1
x

1
cos x sin x
sin x
cos x 
1
x
x0
Now as x  1
sin x
1
x
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A.D.
Notation
sin x
1
x
M R Richards
x 0
as
Is the same as
lim
x 0
sin x
1
x
More generally we write the limit F of a function f(x) at a point x0 as:
f F
as
x  x0
lim f ( x)  F
x  x0
If f  F and g  G as x  x0
Then
af  bg  aF  bG
fg  FG
f
F

g
G
Many limits are trivial e.g.
f(x) = x,
g(x) = x+2
lim f ( x) g ( x)  FG  15
x 3
Limits are non-trivial if
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0 
, ,    ,0 
0 
A.D.
M R Richards
L’Hopital Rule
L’Hopital Rule is used to evaluate 0/0
For the limit lim
x x0
f ( x)
where f(x0) = g(x0) = 0
g ( x)
Then
lim
x  x0
f ( x)
f ' ( x)
 lim
g ( x) x x0 g ' ( x)
Note if f’(x0) and g’(x0) are still zero then:
lim
x  x0
f ' ( x)
f ' ' ( x)
etc
 lim
g ' ( x) x x0 g ' ' ( x)
e.g.
sin x
cos x
 lim
1
x 0
x 0
x
1
lim
e.g.
lim
x 1
x3  x 2  2x  2
As 0/0 use L’Hopital
x3  x2  2
3x 2  2 x  2 3
 lim

x 1
5
3x 2  2 x
e.g.
x3 1
As 0/0 use L’Hopital
x 1 x  1
3x 2
 lim
3
x 1 1
lim
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A.D.
M R Richards
Other Methods can be used 2 such are:
(1)
e.g.
x3 1
lim
x 1 x  1
( x  1)( x 2  x  1)
lim
x 1
x 1
2
lim x  x  1  3
x 1
(2)
e.g.
x3 1
lim
, x  (1  h)
x 1 x  1
1  3h  3h 2  h 3  1
h 0
h
lim 3  3h  h 2  3
lim
h 0
Other examples of tricky limits, not 0/0
e.g.
2x 5  2x  1
x  x 5  x 3  1
lim
Isolating the dominate terms gives:
2
1
 5)
3
x
x
lim
x  5
1
1
x (1  2  5 )
x
x
x 5 (2 
Cancelling
2  2 x 3  x 5
lim
2
x  1  x  2  x  5
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A.D.
M R Richards
e.g.
1
1
1


lim x 2 ( x  1) 2  ( x  1) 2 
x 


Isolate the dominate term
1
1
1
1
 1
1
1 
lim x 2  x 2 (1  ) 2  x 2 (1  ) 2 
x 
x
x 

Factorise
1
1
1
1
1
1  
 
lim x 2  x 2 (1  ) 2  (1  ) 2  
x 
x
x  
 
1
1

1 2
1 2
lim x (1  )  (1  ) 
x 
x
x 

Using the binomial expansion for the expansions:
 (  1) 2  (  1)(  2) 3
(1  t )   1  t 
t 
t ...
2!
3!
1
1
1
1


lim x (1 
 O 2 ...)  (1 
 O 2 ...)
x 
2x
2x
x
x


1 
1
lim x   O 2 ...
x 
x 
x
1
lim 1  O 2 ...  1
x 
x
e.g.
1
1


lim x ( x 2  2) 2  ( x 2  3) 2 
x 


1
1
1
1
 2
2
3 
2
lim x  x 2 (1  2 ) 2  x 2 (1  2 ) 2 
x 
x
x


1
1
3
1 

lim x 2 (1  2  O 4 ...)  (1  2  O 4 ...
x 
x
x
2x
x 

1 5
 5
lim x 2  2  O 4  
x 
x  2
 2x
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O is not zero but it means order, as
coefficient is unimportant.