PPT: Limit, Continuity & Differentiability

# 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
```

## Engineering Mathematics

65 videos|121 docs|94 tests

## FAQs on PPT: Limit, Continuity & Differentiability - Engineering Mathematics - Civil Engineering (CE)

 1. What is the definition of limit in calculus?
Ans. In calculus, the limit of a function is the value that the function approaches as the input approaches a certain point. It represents the behavior of the function near that particular point.
 2. How can we determine if a function is continuous at a specific point?
Ans. A function is said to be continuous at a specific point if three conditions are met: the function is defined at that point, the limit of the function as it approaches that point exists, and the value of the function at that point is equal to the limit.
 3. What is the difference between differentiability and continuity?
Ans. Continuity refers to the smoothness of a function, where there are no gaps, jumps, or breaks. Differentiability, on the other hand, refers to the existence of the derivative of a function at a particular point. A function can be continuous but not differentiable, but if a function is differentiable, it is also continuous.
 4. How do we calculate the derivative of a function?
Ans. The derivative of a function can be calculated using various rules and formulas, such as the power rule, product rule, quotient rule, and chain rule. These rules allow us to find the rate at which the function is changing with respect to its input.
 5. Can a function be continuous but not differentiable?
Ans. Yes, a function can be continuous but not differentiable. This occurs when there is a sharp corner or vertical tangent at a certain point, where the slope of the function is not defined. Examples of such functions include absolute value functions at their vertex and functions with cusp points.

## Engineering Mathematics

65 videos|121 docs|94 tests

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