PPT: Limit, Continuity & Differentiability | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


Functions
? A function is a special relationship where each input has a single output.
? It is often written as "f(x)" where x is the input value.
? e.g. - f(x) = x/2 is a function, because each input "x" has a single output "x/2“
f(8) = 8/2 = 4,  f(-24) = -12
? Explicit function : An explicit function is one which is given in terms of 
the independent variable. i.e. z = f(x
1
, x
2
,….,x
n
)
e.g. y = x
2
+ 3x – 8
? Implicit functions : on the other hand, are usually given in terms of both dependent and 
independent variables. i.e. ?(z, x
1
, x
2
,….,x
n
) = 0     e.g. y + x
2
- 3x + 8 = 0
Dependent 
Variable
Independent 
Variables
Page 2


Functions
? A function is a special relationship where each input has a single output.
? It is often written as "f(x)" where x is the input value.
? e.g. - f(x) = x/2 is a function, because each input "x" has a single output "x/2“
f(8) = 8/2 = 4,  f(-24) = -12
? Explicit function : An explicit function is one which is given in terms of 
the independent variable. i.e. z = f(x
1
, x
2
,….,x
n
)
e.g. y = x
2
+ 3x – 8
? Implicit functions : on the other hand, are usually given in terms of both dependent and 
independent variables. i.e. ?(z, x
1
, x
2
,….,x
n
) = 0     e.g. y + x
2
- 3x + 8 = 0
Dependent 
Variable
Independent 
Variables
Functions
? Composite function : z = f(x, y) where, x = ?(t) and y = ?(t)
Some special functions
? Even function : f(-x) = f(x)  e.g. – cos x, x
2
? Odd function : f(-x) = - f(x)
? Modulus function : f(x) = |x|= x ; x > 0
= -x ; x < 0
= 0 ; x = 0
? Greatest integer function : For all real numbers, x, the greatest integer function returns the 
largest integer less than or equal to x. f(x) = [x] = n ? z |where n = x < n+1.
e.g. [7.2] = 7
Page 3


Functions
? A function is a special relationship where each input has a single output.
? It is often written as "f(x)" where x is the input value.
? e.g. - f(x) = x/2 is a function, because each input "x" has a single output "x/2“
f(8) = 8/2 = 4,  f(-24) = -12
? Explicit function : An explicit function is one which is given in terms of 
the independent variable. i.e. z = f(x
1
, x
2
,….,x
n
)
e.g. y = x
2
+ 3x – 8
? Implicit functions : on the other hand, are usually given in terms of both dependent and 
independent variables. i.e. ?(z, x
1
, x
2
,….,x
n
) = 0     e.g. y + x
2
- 3x + 8 = 0
Dependent 
Variable
Independent 
Variables
Functions
? Composite function : z = f(x, y) where, x = ?(t) and y = ?(t)
Some special functions
? Even function : f(-x) = f(x)  e.g. – cos x, x
2
? Odd function : f(-x) = - f(x)
? Modulus function : f(x) = |x|= x ; x > 0
= -x ; x < 0
= 0 ; x = 0
? Greatest integer function : For all real numbers, x, the greatest integer function returns the 
largest integer less than or equal to x. f(x) = [x] = n ? z |where n = x < n+1.
e.g. [7.2] = 7
Functions
? Symmetric properties of the curve:
Let f( x, y) = c be the equation of the curve
1) If f( x, y) contains only even power of x i.e. f(-x, y) = f( x, y) then it is symmetric about y axis
2) If f( x, y) contains only even powers of y i.e. f( x, -y) = f( x, y) then it is symmetric about x 
axis
3) If f( x, y) = f ( y, x) then the curve is symmetric about y = x
Page 4


Functions
? A function is a special relationship where each input has a single output.
? It is often written as "f(x)" where x is the input value.
? e.g. - f(x) = x/2 is a function, because each input "x" has a single output "x/2“
f(8) = 8/2 = 4,  f(-24) = -12
? Explicit function : An explicit function is one which is given in terms of 
the independent variable. i.e. z = f(x
1
, x
2
,….,x
n
)
e.g. y = x
2
+ 3x – 8
? Implicit functions : on the other hand, are usually given in terms of both dependent and 
independent variables. i.e. ?(z, x
1
, x
2
,….,x
n
) = 0     e.g. y + x
2
- 3x + 8 = 0
Dependent 
Variable
Independent 
Variables
Functions
? Composite function : z = f(x, y) where, x = ?(t) and y = ?(t)
Some special functions
? Even function : f(-x) = f(x)  e.g. – cos x, x
2
? Odd function : f(-x) = - f(x)
? Modulus function : f(x) = |x|= x ; x > 0
= -x ; x < 0
= 0 ; x = 0
? Greatest integer function : For all real numbers, x, the greatest integer function returns the 
largest integer less than or equal to x. f(x) = [x] = n ? z |where n = x < n+1.
e.g. [7.2] = 7
Functions
? Symmetric properties of the curve:
Let f( x, y) = c be the equation of the curve
1) If f( x, y) contains only even power of x i.e. f(-x, y) = f( x, y) then it is symmetric about y axis
2) If f( x, y) contains only even powers of y i.e. f( x, -y) = f( x, y) then it is symmetric about x 
axis
3) If f( x, y) = f ( y, x) then the curve is symmetric about y = x
Limits
? Limit of a function: let f(x) be defined in neighbourhood of a ? R, then l ? R is said to be limit 
f(x) as x approaches a if for given ? > 0 & d > 0 such that |f(x) – l|< ? whenever |x – a|< d.
? li m
?? ? ?? ?? ( ?? ) = l 
? Left limit : when x < a, li m
?? ? ?? -
?? ( ?? ) = li m
h ? 0
?? ( ?? - h ) where h = a - x
? Right limit : when x < a, li m
?? ? ?? +
?? ( ?? ) = li m
h ? 0
?? ( ?? + h ) where h = x – a
? Limit exists only if li m
?? ? ?? -
?? ( ?? ) = li m
?? ? ?? +
?? ( ?? )
? L’ Hospital’s rule :  li m
?? ? ?? ?? ( ?? )
?? ( ?? )
= li m
?? ? ?? ?? '
( ?? )
?? '
( ?? )
[ as 
0
0
or 
8
8
]
Page 5


Functions
? A function is a special relationship where each input has a single output.
? It is often written as "f(x)" where x is the input value.
? e.g. - f(x) = x/2 is a function, because each input "x" has a single output "x/2“
f(8) = 8/2 = 4,  f(-24) = -12
? Explicit function : An explicit function is one which is given in terms of 
the independent variable. i.e. z = f(x
1
, x
2
,….,x
n
)
e.g. y = x
2
+ 3x – 8
? Implicit functions : on the other hand, are usually given in terms of both dependent and 
independent variables. i.e. ?(z, x
1
, x
2
,….,x
n
) = 0     e.g. y + x
2
- 3x + 8 = 0
Dependent 
Variable
Independent 
Variables
Functions
? Composite function : z = f(x, y) where, x = ?(t) and y = ?(t)
Some special functions
? Even function : f(-x) = f(x)  e.g. – cos x, x
2
? Odd function : f(-x) = - f(x)
? Modulus function : f(x) = |x|= x ; x > 0
= -x ; x < 0
= 0 ; x = 0
? Greatest integer function : For all real numbers, x, the greatest integer function returns the 
largest integer less than or equal to x. f(x) = [x] = n ? z |where n = x < n+1.
e.g. [7.2] = 7
Functions
? Symmetric properties of the curve:
Let f( x, y) = c be the equation of the curve
1) If f( x, y) contains only even power of x i.e. f(-x, y) = f( x, y) then it is symmetric about y axis
2) If f( x, y) contains only even powers of y i.e. f( x, -y) = f( x, y) then it is symmetric about x 
axis
3) If f( x, y) = f ( y, x) then the curve is symmetric about y = x
Limits
? Limit of a function: let f(x) be defined in neighbourhood of a ? R, then l ? R is said to be limit 
f(x) as x approaches a if for given ? > 0 & d > 0 such that |f(x) – l|< ? whenever |x – a|< d.
? li m
?? ? ?? ?? ( ?? ) = l 
? Left limit : when x < a, li m
?? ? ?? -
?? ( ?? ) = li m
h ? 0
?? ( ?? - h ) where h = a - x
? Right limit : when x < a, li m
?? ? ?? +
?? ( ?? ) = li m
h ? 0
?? ( ?? + h ) where h = x – a
? Limit exists only if li m
?? ? ?? -
?? ( ?? ) = li m
?? ? ?? +
?? ( ?? )
? L’ Hospital’s rule :  li m
?? ? ?? ?? ( ?? )
?? ( ?? )
= li m
?? ? ?? ?? '
( ?? )
?? '
( ?? )
[ as 
0
0
or 
8
8
]
Limits
? Example : Applying L’ Hospitals' rule
li m
?? ? 0
1 - cos 3??
?? si n 2??
[
0
0
]
li m
?? ? 0
3 si n 3??
si n 2?? + 2?? cos 2??
li m
?? ? 0
9 cos 3??
2 ?? ????2 ?? + 2 cos 2 ?? - 4?? si n 2??
= 
9
4
IMP