CALCULUS

Differential Calculus
Integral Calculus
Differential Equations
CALCULUS
Differential Calculus

Limits

A limit is the value that a function (or

Right-sided Limits: Let f (x) be a function
sequence) approaches as the input (or
index) approaches some value. defined at all values in an open interval of
the form (a ,c ), and let L be a real number.
If the values of the function approach L as
Intuitive Definition of a Limit
the values of x| x > a approach the number a
, then we say that L is the limit of f (x) as x
Let f (x) be a function defined at all values approaches a from the right, or
in an open interval containing a , with the
possible exception of a itself, and let L be a lim ¿.
+¿
x→ a f ( x ) =L¿
real number. If all values of the function
f (x) approaches the number L as the
values of x| x ≠ a approach the number a, Relation of Two-sided and One-sided
then we say that the limit of f (x) as x Limits
approaches a is L. Symbolically, we can
express this idea as: Let f (x) be a function defined at all values
lim f ( x )=L. in an open interval containing a , with the
x →a
possible exception of a itself, and let L be a
real number. Then,
Formal Definition of a Limit
(Delta-Epsilon Proof) lim f ( x )=L ,iff lim ¿
−¿
x →a x→ a f ( x )= L∧ lim ¿¿
+¿
x→a f ( x )=L ¿

Let c and L be real numbers. The function

f (x) has a limit L as x approaches to c if,
Infinite Limits
given any positive number ε , there exist a
positive number δ such that for all x ,
0<| x−c|<δ ⟹|f ( x )−L|< ε Infinite Limits from the Left
Then, lim ¿
−¿
x→ a f ( x )=+∞ lim ¿¿
lim f ( x )=L.
−¿
x→a f (x ) =−∞¿

x →c
Infinite Limits from the Left
Two Important Limits lim ¿
+¿
x→ a f ( x ) =+ ∞ +¿
lim ¿¿
x→ a f ( x )=−∞¿

Let a be a real number and c be a constant. Two-sided Infinite Limits

lim x=a lim f ( x )=± ∞
x →a
x →a
iff lim ¿
−¿
lim c=c x → a f ( x ) =±∞ ∧ lim
+¿
x→ a f ( x )=± ∞ ¿
¿¿
x →a

Limit Laws
One-sided Limits

Let f (x) and g(x ) be defined for all x ≠ a

Left-sided Limits: Let f (x) be a function over some open interval containing a .
defined at all values in an open interval of Assume that L and M are real numbers
the form z , and let L be a real number. If the f ( x )=L and lim g ( x )=M . Let
such that lim
values of the function approach L as the x →a x →a

values of x| x < a approach the number a , c be a constant. Then each of the following
statement holds:
then we say that L is the limit of f (x) as x
approaches a from the left, or lim ( f ( x ) + g ( x ))=lim f ( x )+ ¿ lim g ( x )=L+ M ¿
x →a x→ a x→a
lim ¿.
−¿
x→ a f ( x )= L¿

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Differential Calculus
lim ( f ( x ) −g ( x ) )=lim f ( x )−¿ lim g ( x )= L−M ¿
x →a x →a x →a
Let f ( x ) , g (x) and h( x) be defined for all x ≠ a
lim c ( f ( x ) )=c lim f ( x ) =cL over an open interval containing a . If
x →a x→ a f (x)≤ g (x)≤ h(x) for all x ≠ a in an open
lim ( f ( x ) ∙ g ( x ) )=lim f ( x ) ∙ lim g ( x )=L ∙ M interval containing a and
x →a x→ a x→ a
lim f ( x )=L=lim h ( x )
lim f ( x ) x →a x→a
f ( x ) x →a L
lim = = , M≠0 Where L is a real number, then
x →a g(x ) lim g ( x ) M
x→a
lim g ( x )=L .
x →a

n n n + ¿¿
lim ( f ( x ) ) =(lim f ( x )) =L , n ∈ Z
x →a x→ a

√
lim √ f (x )= n lim f ( x ) =√ L ¿
n n

x →a x→ a
Important Limits

Limits of Polynomial and Rational lim sin θ=0

θ→0
Functions
lim cos θ=1
θ→0
Let p(x )and q (x) be polynomial functions.
Let a be a real number. Then, sin θ
lim =1
θ→0 θ
lim p ( x )= p (a)
x →a
tan θ
p ( x ) p(a) lim =1
lim = , q(a)≠ 0. θ→0 θ
x →a q (x) q(a)
−1
sin θ
If for all x ≠ a , p ( x )=q (x) over some open lim =1
θ→0 θ
interval containing a , then,
−1
lim p ( x )=lim q ( x ) . tan θ
lim =1
x →a x→a θ→0 θ
1−cos θ
Limits of Rational Functions having the lim =0
θ→0 θ
Indeterminate Form, 0/0
1−cos θ
lim =0
Methods in Calculating a Limit with the θ→0 θ
Indeterminate Form, 0/0
ln ⁡(1+ x)
lim =1
Given a function h ( x )=f ( x) /g( x) where x →0 x
x ≠ a|a ∈( j , k ), and f (x) and g(x ) is equal lim ¿ ¿
to zero, x→ ∞
1. Factor f (x) and g(x ) and cancel out any
common factors.
lim ¿ ¿
x→ ∞
2. If the numerator or denominator contains
a sum or difference involving a square root, lim ¿ ¿
x→ ∞
multiply both the numerator and
denominator by the conjugate of the x
expression involving the square root.
lim a =1
x →0
3. If f (x)/ g(x ) is a complex fraction,
simplify the expression by doing algebraic
manipulation. Continuity at a Point

The Squeeze Theorem

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Differential Calculus
A function is continuous at a point a , if and only
if the following three conditions are satisfied: The Intermediate Value Theorem
1. f (a) is defined
Let f (x) be continuous over a closed
2. lim f (x ) exists bounded interval [ a , b ]. If z is any real
x →a
number between f (a) and f (b), then there
3. lim f (x )=f (a) is a number c in [ a , b ] satisfying f ( c ) =z.
x →a

A function is discontinuous at point a if it fails

one of the three conditions. L’Hôpital’s Rule for Indeterminate Cases

Suppose f and g are differentiable functions

over an open interval containing a , except
possibly at a . If lim f (x )=lim g(x)=0 or
x →a x →a
lim f (x )=lim g(x)=± ∞, then,
x →a x →a
f (x) f ' ( x)
lim =lim ,
x →a g(x ) x→ a g '(x )

where f ' (x) and g '(x ) are the derivative of

the functions f and g.
Types of Discontinuity

Removable Discontinuity
f (x) has a removable discontinuity at a if
lim f (x )=L and f ( x ) ≠ L .
x →a Limits at Infinity

Jump Discontinuity
Functions with Limits at Infinity
f (x) has a jump discontinuity at a if both
lim ¿ and lim ¿ are real numbers,
+¿
x→ a f (x)¿ x→ a f (x)¿
−¿
lim ¿ lim ¿
lim ¿ ≠ lim ¿ . −¿
x→ 0
1
=−∞ ¿
and x→ 0
+¿ 1
=+∞ ¿
but +¿
x→ a f (x)¿
−¿
x → a f (x)¿
x x
1 1
lim =0and lim =0
Infinite Discontinuity x→−∞ x x→+∞ x

f (x) has a jump discontinuity at a if

lim ¿ or lim ¿. Limits at Infinity for Rational Functions
+¿ −¿
x→ a f (x)=± ∞ ¿ x→ a f ( x )=± ∞ ¿
1. If the degrees of the polynomials are
equal, then the limit is the ratio of the
Continuity over an Interval leading coefficients.
2. If the degree of the polynomial in the
A function is said to be continuous from the denominator is greater than the degree of
right at a if lim ¿ the polynomial in the numerator, then the
+¿
x→ a f (x)=f ( a ) .¿
limit is zero.
A function is said to be continuous from the 3. If the degree of the polynomial in the
left at a if lim ¿ denominator is less than the degree of the
−¿
x→ a f (x)=f ( a ) .¿
polynomial in the numerator, then the limit
approaches positive or negative infinity.
Composite Function Theorem

Derivatives
If f (x) is continuous at L and lim g (x)=L,
x →a
then,
Slope of a Secant Line
(
lim f (g ( x ) )=f lim g ( x ) =f (L).
x →a x →a )
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CALCULUS
Differential Calculus
Let f be a function defined on an interval I d
( c )=0
containing a . If x ≠ a is in I , then, dx
f ( x )−f ( a) The Power Rule
Q= d n
x−a ( u ) =n un−1 u '
Is a difference quotient. dx
Constant Multiple Rule
If h ≠ 0 and a+ h is in I , then d
( cu )=cu '
f ( a+h ) −f (a) dx
Q= Sum and Difference Rule
h
is a difference quotient with increment h . d
( u ± v )=u ' ± v '
dx
Product Rule
Slope of a Tangent Line
d
( uv )=u v ' + v u '
dx
Let f be a function defined on an interval I Quotient Rule
containing a . The tangent line to f (x) at a is
()
' '
d u v u −u v
the line passing through the point (a , f ( a ) ) = 2
dx v v
having the slope,
f ( x )−f (a)
mtan =lim Chain Rule
x→ a x−a
Provided that the limit exists.
Given the functions u and v , the derivative
of the composite function v (u ( x ) ) is
d ' '
(v ( u ( x ) ) )=v ( u ( x ) ) u ( x ) .
dx
If v is a function of y then,
du
Alternatively, using the increment h such dx
that h ≠ 0 and a+ h is in I
f ( a+h )−f (a)
mtan =lim
h→0 h

The Derivative of a Function at a Point

Let f (x) be a function defined in an open

interval containing a . The derivative of the
function f ( x ) at a denoted by f ' (a) is
defined by
f ( x )−f (a)
f ' (a)=lim
x→a x −a
provided this limit exists.

Alternatively, using the increment h ,

f ( a+ h )−f (a)
f ' (a)=lim
h →0 h

Basic Differentiation Rules

Given the functions u and v , and real

constants c and n ,

The Constant Rule

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Differential Calculus

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Differential Calculus

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Differential Calculus

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CALCULUS
Differential Calculus