Essential Calculus
Limits of Functions, RL1.1.1
Dr Mayank Goel
Department of Mathematics
BITS Pilani KK Birla Goa Campus
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Outline
1
Introduction to Limits
2
Limit Definition
3
Evaluation of Limits
4
Limit Principles
5
Examples
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
Sometimes we can not find the function value at a point directly but it
is useful to see what happens to the function when we move closer to
that point.
Let’s consider an example f (x) =
x2 − 1
.
x −1
(x − 1)(x + 1)
= x + 1.
x −1
0×1
0
But for x = 1 we can not do the same as
= is undetermined.
0
0
So, instead of trying to work it out for x = 1, lets try approaching it
closer and closer to 1
For x 6= 1, we can easily write f (x) =
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
Sometimes we can not find the function value at a point directly but it
is useful to see what happens to the function when we move closer to
that point.
Let’s consider an example f (x) =
x2 − 1
.
x −1
(x − 1)(x + 1)
= x + 1.
x −1
0×1
0
But for x = 1 we can not do the same as
= is undetermined.
0
0
So, instead of trying to work it out for x = 1, lets try approaching it
closer and closer to 1
For x 6= 1, we can easily write f (x) =
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
Sometimes we can not find the function value at a point directly but it
is useful to see what happens to the function when we move closer to
that point.
Let’s consider an example f (x) =
x2 − 1
.
x −1
(x − 1)(x + 1)
= x + 1.
x −1
0×1
0
= is undetermined.
But for x = 1 we can not do the same as
0
0
So, instead of trying to work it out for x = 1, lets try approaching it
closer and closer to 1
For x 6= 1, we can easily write f (x) =
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
Sometimes we can not find the function value at a point directly but it
is useful to see what happens to the function when we move closer to
that point.
Let’s consider an example f (x) =
x2 − 1
.
x −1
(x − 1)(x + 1)
= x + 1.
x −1
0×1
0
= is undetermined.
But for x = 1 we can not do the same as
0
0
So, instead of trying to work it out for x = 1, lets try approaching it
closer and closer to 1
For x 6= 1, we can easily write f (x) =
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
Sometimes we can not find the function value at a point directly but it
is useful to see what happens to the function when we move closer to
that point.
Let’s consider an example f (x) =
x2 − 1
.
x −1
(x − 1)(x + 1)
= x + 1.
x −1
0×1
0
= is undetermined.
But for x = 1 we can not do the same as
0
0
So, instead of trying to work it out for x = 1, lets try approaching it
closer and closer to 1
For x 6= 1, we can easily write f (x) =
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Motivation: Behavior of the function
for increasing values of x to 1, function f (x) gives the following result
Observation
x
0.3
0.6
0.9
0.99
0.999
···
x2 − 1
gets closer to 2 as x gets
x −1
closer to 1.
x 2 −1
x−1
1.3
1.6
1.9
1.99
1.999
···
Conclusion
We can not give value exactly at
x = 1 but as we get closer and
closer to x = 1 the answer get
closer and closer to 2.
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
So, when we ignore what happen to the function f at x = x0 but if f
approaches to some number L as we get closer and closer to x = x0
is represented by the word ”Limit”. And is written as
lim f (x) = L
x→x0
Question: Testing from the one side is not sufficient (why ?)
In previous example
x
1.5
1.2
1.1
1.01
1.001
···
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
2.5
2.2
2.1
2.01
2.001
···
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
So, when we ignore what happen to the function f at x = x0 but if f
approaches to some number L as we get closer and closer to x = x0
is represented by the word ”Limit”. And is written as
lim f (x) = L
x→x0
Question: Testing from the one side is not sufficient (why ?)
In previous example
x
1.5
1.2
1.1
1.01
1.001
···
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
2.5
2.2
2.1
2.01
2.001
···
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
So, when we ignore what happen to the function f at x = x0 but if f
approaches to some number L as we get closer and closer to x = x0
is represented by the word ”Limit”. And is written as
lim f (x) = L
x→x0
Question: Testing from the one side is not sufficient (why ?)
In previous example
x
1.5
1.2
1.1
1.01
1.001
···
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
2.5
2.2
2.1
2.01
2.001
···
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
So, when we ignore what happen to the function f at x = x0 but if f
approaches to some number L as we get closer and closer to x = x0
is represented by the word ”Limit”. And is written as
lim f (x) = L
x→x0
Question: Testing from the one side is not sufficient (why ?)
In previous example
x
1.5
1.2
1.1
1.01
1.001
···
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
2.5
2.2
2.1
2.01
2.001
···
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
So, when we ignore what happen to the function f at x = x0 but if f
approaches to some number L as we get closer and closer to x = x0
is represented by the word ”Limit”. And is written as
lim f (x) = L
x→x0
Question: Testing from the one side is not sufficient (why ?)
In previous example
x
1.5
1.2
1.1
1.01
1.001
···
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
2.5
2.2
2.1
2.01
2.001
···
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
So, when we ignore what happen to the function f at x = x0 but if f
approaches to some number L as we get closer and closer to x = x0
is represented by the word ”Limit”. And is written as
lim f (x) = L
x→x0
Question: Testing from the one side is not sufficient (why ?)
In previous example
x
1.5
1.2
1.1
1.01
1.001
···
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
2.5
2.2
2.1
2.01
2.001
···
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
So, when we ignore what happen to the function f at x = x0 but if f
approaches to some number L as we get closer and closer to x = x0
is represented by the word ”Limit”. And is written as
lim f (x) = L
x→x0
Question: Testing from the one side is not sufficient (why ?)
In previous example
x
1.5
1.2
1.1
1.01
1.001
···
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
2.5
2.2
2.1
2.01
2.001
···
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
So, when we ignore what happen to the function f at x = x0 but if f
approaches to some number L as we get closer and closer to x = x0
is represented by the word ”Limit”. And is written as
lim f (x) = L
x→x0
Question: Testing from the one side is not sufficient (why ?)
In previous example
x
1.5
1.2
1.1
1.01
1.001
···
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
2.5
2.2
2.1
2.01
2.001
···
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
To calculate limx→x0 f (x)
Limit approaching from left to right is known as left hand limit
(LHL) and is expressed as
limx→x − f (x).
0
Limit approaching from right to left is called right hand limit
(RHL) and is expressed as
limx→x0+ f (x).
We say limx→x0 f (x) exists and is equal to L iff
RHL = LHL = L, where L is a finite number.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
To calculate limx→x0 f (x)
Limit approaching from left to right is known as left hand limit
(LHL) and is expressed as
limx→x − f (x).
0
Limit approaching from right to left is called right hand limit
(RHL) and is expressed as
limx→x0+ f (x).
We say limx→x0 f (x) exists and is equal to L iff
RHL = LHL = L, where L is a finite number.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
To calculate limx→x0 f (x)
Limit approaching from left to right is known as left hand limit
(LHL) and is expressed as
limx→x − f (x).
0
Limit approaching from right to left is called right hand limit
(RHL) and is expressed as
limx→x0+ f (x).
We say limx→x0 f (x) exists and is equal to L iff
RHL = LHL = L, where L is a finite number.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Left Hand Limit and Right Hand Limit
To calculate limx→x0 f (x)
Limit approaching from left to right is known as left hand limit
(LHL) and is expressed as
limx→x − f (x).
0
Limit approaching from right to left is called right hand limit
(RHL) and is expressed as
limx→x0+ f (x).
We say limx→x0 f (x) exists and is equal to L iff
RHL = LHL = L, where L is a finite number.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Question: Can we always evaluate the limits of a function using
calculator as done in previous example ?
Answer is No.
Consider a function f (x) = sin πx , x 6= 0, where x is in radian.
RHL
x
0.1
0.01
0.001
0.0001
···
f (x)
0
0
0
0
···
LHL
x
-0.1
-0.01
-0.001
-0.0001
···
f (x)
0
0
0
0
···
It appears that limx→0 sin πx should be 0 but in fact it is not true.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
To calculate LHL of a function f as x tends to a
lim f (x) = lim f (x0 − h)
x→x0−
h→0
To calculate RHL of a function f as x tends to a
lim f (x) = lim f (x0 + h)
x→x0+
h→0
Justification ?
Let’s consider previous example again i.e. limx→1
LHL = limx→1− f (x) = limh→0 f (1 − h)
= limh→0
(1−h)2 −1
(1−h)−1
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
To calculate LHL of a function f as x tends to a
lim f (x) = lim f (x0 − h)
x→x0−
h→0
To calculate RHL of a function f as x tends to a
lim f (x) = lim f (x0 + h)
x→x0+
h→0
Justification ?
Let’s consider previous example again i.e. limx→1
LHL = limx→1− f (x) = limh→0 f (1 − h)
= limh→0
(1−h)2 −1
(1−h)−1
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
To calculate LHL of a function f as x tends to a
lim f (x) = lim f (x0 − h)
x→x0−
h→0
To calculate RHL of a function f as x tends to a
lim f (x) = lim f (x0 + h)
x→x0+
h→0
Justification ?
Let’s consider previous example again i.e. limx→1
LHL = limx→1− f (x) = limh→0 f (1 − h)
= limh→0
(1−h)2 −1
(1−h)−1
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
To calculate LHL of a function f as x tends to a
lim f (x) = lim f (x0 − h)
x→x0−
h→0
To calculate RHL of a function f as x tends to a
lim f (x) = lim f (x0 + h)
x→x0+
h→0
Justification ?
Let’s consider previous example again i.e. limx→1
LHL = limx→1− f (x) = limh→0 f (1 − h)
= limh→0
(1−h)2 −1
(1−h)−1
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
To calculate LHL of a function f as x tends to a
lim f (x) = lim f (x0 − h)
x→x0−
h→0
To calculate RHL of a function f as x tends to a
lim f (x) = lim f (x0 + h)
x→x0+
h→0
Justification ?
Let’s consider previous example again i.e. limx→1
LHL = limx→1− f (x) = limh→0 f (1 − h)
= limh→0
(1−h)2 −1
(1−h)−1
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
To calculate LHL of a function f as x tends to a
lim f (x) = lim f (x0 − h)
x→x0−
h→0
To calculate RHL of a function f as x tends to a
lim f (x) = lim f (x0 + h)
x→x0+
h→0
Justification ?
Let’s consider previous example again i.e. limx→1
LHL = limx→1− f (x) = limh→0 f (1 − h)
= limh→0
(1−h)2 −1
(1−h)−1
Dr Mayank Goel
Essential Calculus
x 2 −1
x−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Now since h 6= 0 and hence (1 − h) − 1 6= 0, so it is totaly right to
rewrite the above as
limh→0
((1 − h) − 1)((1 − h) + 1)
= lim ((1 − h) + 1),
h→0
(1 − h) − 1
Finally, taking the limit h → 0, we have LHL = 2
Similarly, RHL = limx→1+ f (x) = limh→0 f (1 + h) = limh→0
= limh→0 ((1 + h) + 1) = 2,
Since, LHL = RHL = 2, hence limx→1
Dr Mayank Goel
x 2 −1
x−1
=2
Essential Calculus
(1+h)2 −1
(1+h)−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Now since h 6= 0 and hence (1 − h) − 1 6= 0, so it is totaly right to
rewrite the above as
limh→0
((1 − h) − 1)((1 − h) + 1)
= lim ((1 − h) + 1),
h→0
(1 − h) − 1
Finally, taking the limit h → 0, we have LHL = 2
Similarly, RHL = limx→1+ f (x) = limh→0 f (1 + h) = limh→0
= limh→0 ((1 + h) + 1) = 2,
Since, LHL = RHL = 2, hence limx→1
Dr Mayank Goel
x 2 −1
x−1
=2
Essential Calculus
(1+h)2 −1
(1+h)−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Now since h 6= 0 and hence (1 − h) − 1 6= 0, so it is totaly right to
rewrite the above as
limh→0
((1 − h) − 1)((1 − h) + 1)
= lim ((1 − h) + 1),
h→0
(1 − h) − 1
Finally, taking the limit h → 0, we have LHL = 2
Similarly, RHL = limx→1+ f (x) = limh→0 f (1 + h) = limh→0
= limh→0 ((1 + h) + 1) = 2,
Since, LHL = RHL = 2, hence limx→1
Dr Mayank Goel
x 2 −1
x−1
=2
Essential Calculus
(1+h)2 −1
(1+h)−1
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Example
Find limit as x → 0 of
f (x) =
sin x
,
x
where x is in radian.
RHL of a function f as x tends to 0 is
lim f (x) = lim f (0 + h)
x→0+
h→0
sin h
sin (0 + h)
= lim
h→0
h→0 h
0+h
= lim
= lim
h→0
1
[h − h3 /3! + h5 /5! − · · · ] = lim [1 − h2 /3! + h4 /5! − · · · ] = 1
h→0
h
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Example
Find limit as x → 0 of
f (x) =
sin x
,
x
where x is in radian.
RHL of a function f as x tends to 0 is
lim f (x) = lim f (0 + h)
x→0+
h→0
sin h
sin (0 + h)
= lim
h→0
h→0 h
0+h
= lim
= lim
h→0
1
[h − h3 /3! + h5 /5! − · · · ] = lim [1 − h2 /3! + h4 /5! − · · · ] = 1
h→0
h
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Example
Find limit as x → 0 of
f (x) =
sin x
,
x
where x is in radian.
RHL of a function f as x tends to 0 is
lim f (x) = lim f (0 + h)
x→0+
h→0
sin h
sin (0 + h)
= lim
h→0
h→0 h
0+h
= lim
= lim
h→0
1
[h − h3 /3! + h5 /5! − · · · ] = lim [1 − h2 /3! + h4 /5! − · · · ] = 1
h→0
h
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Example
Find limit as x → 0 of
f (x) =
sin x
,
x
where x is in radian.
RHL of a function f as x tends to 0 is
lim f (x) = lim f (0 + h)
x→0+
h→0
sin h
sin (0 + h)
= lim
h→0
h→0 h
0+h
= lim
= lim
h→0
1
[h − h3 /3! + h5 /5! − · · · ] = lim [1 − h2 /3! + h4 /5! − · · · ] = 1
h→0
h
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Example
Find limit as x → 0 of
f (x) =
sin x
,
x
where x is in radian.
RHL of a function f as x tends to 0 is
lim f (x) = lim f (0 + h)
x→0+
h→0
sin h
sin (0 + h)
= lim
h→0
h→0 h
0+h
= lim
= lim
h→0
1
[h − h3 /3! + h5 /5! − · · · ] = lim [1 − h2 /3! + h4 /5! − · · · ] = 1
h→0
h
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Example
Find limit as x → 0 of
f (x) =
sin x
,
x
where x is in radian.
RHL of a function f as x tends to 0 is
lim f (x) = lim f (0 + h)
x→0+
h→0
sin h
sin (0 + h)
= lim
h→0
h→0 h
0+h
= lim
= lim
h→0
1
[h − h3 /3! + h5 /5! − · · · ] = lim [1 − h2 /3! + h4 /5! − · · · ] = 1
h→0
h
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Example
Find limit as x → 0 of
f (x) =
sin x
,
x
where x is in radian.
RHL of a function f as x tends to 0 is
lim f (x) = lim f (0 + h)
x→0+
h→0
sin h
sin (0 + h)
= lim
h→0
h→0 h
0+h
= lim
= lim
h→0
1
[h − h3 /3! + h5 /5! − · · · ] = lim [1 − h2 /3! + h4 /5! − · · · ] = 1
h→0
h
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Similarly, LHL
lim
h→0
sin (0 − h)
− sin h
= lim
=1
h→0
0−h
−h
Since LHL = RHL and is finite, so limit exits and it is 1, i.e.
lim
x→0
sin x
= 1.
x
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Similarly, LHL
lim
h→0
sin (0 − h)
− sin h
= lim
=1
h→0
0−h
−h
Since LHL = RHL and is finite, so limit exits and it is 1, i.e.
lim
x→0
sin x
= 1.
x
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Similarly, LHL
lim
h→0
sin (0 − h)
− sin h
= lim
=1
h→0
0−h
−h
Since LHL = RHL and is finite, so limit exits and it is 1, i.e.
lim
x→0
sin x
= 1.
x
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Similarly, LHL
lim
h→0
sin (0 − h)
− sin h
= lim
=1
h→0
0−h
−h
Since LHL = RHL and is finite, so limit exits and it is 1, i.e.
lim
x→0
sin x
= 1.
x
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Evaluation of Limits
Similarly, LHL
lim
h→0
sin (0 − h)
− sin h
= lim
=1
h→0
0−h
−h
Since LHL = RHL and is finite, so limit exits and it is 1, i.e.
lim
x→0
sin x
= 1.
x
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Example
Find limx→0
sin x
.
|x|
(
Function can be rewritten as f (x) =
sin x
−x ,
sin x
x ,
x <0
x >0
LHL
lim f (x) = lim f (0 − h)
x→0−
h→0
= lim
h→0
= lim
h→0
sin (0 − h)
−(0 − h)
− sin h
= −1
h
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Example
Find limx→0
sin x
.
|x|
(
Function can be rewritten as f (x) =
sin x
−x ,
sin x
x ,
x <0
x >0
LHL
lim f (x) = lim f (0 − h)
x→0−
h→0
= lim
h→0
= lim
h→0
sin (0 − h)
−(0 − h)
− sin h
= −1
h
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Example
Find limx→0
sin x
.
|x|
(
Function can be rewritten as f (x) =
sin x
−x ,
sin x
x ,
x <0
x >0
LHL
lim f (x) = lim f (0 − h)
x→0−
h→0
= lim
h→0
= lim
h→0
sin (0 − h)
−(0 − h)
− sin h
= −1
h
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Example
Find limx→0
sin x
.
|x|
(
Function can be rewritten as f (x) =
sin x
−x ,
sin x
x ,
x <0
x >0
LHL
lim f (x) = lim f (0 − h)
x→0−
h→0
= lim
h→0
= lim
h→0
sin (0 − h)
−(0 − h)
− sin h
= −1
h
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Example
Find limx→0
sin x
.
|x|
(
Function can be rewritten as f (x) =
sin x
−x ,
sin x
x ,
x <0
x >0
LHL
lim f (x) = lim f (0 − h)
x→0−
h→0
= lim
h→0
= lim
h→0
sin (0 − h)
−(0 − h)
− sin h
= −1
h
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Similarly, RHL
lim f (x) = lim f (0 + h) =
x→0+
h→0
sin (0 + h)
(0 + h)
sin h
=1
h→0 h
= lim
Since LHL 6= RHL, so limit does not exits.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Similarly, RHL
lim f (x) = lim f (0 + h) =
x→0+
h→0
sin (0 + h)
(0 + h)
sin h
=1
h→0 h
= lim
Since LHL 6= RHL, so limit does not exits.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Similarly, RHL
lim f (x) = lim f (0 + h) =
x→0+
h→0
sin (0 + h)
(0 + h)
sin h
=1
h→0 h
= lim
Since LHL 6= RHL, so limit does not exits.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Similarly, RHL
lim f (x) = lim f (0 + h) =
x→0+
h→0
sin (0 + h)
(0 + h)
sin h
=1
h→0 h
= lim
Since LHL 6= RHL, so limit does not exits.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Evaluation of Limits
Similarly, RHL
lim f (x) = lim f (0 + h) =
x→0+
h→0
sin (0 + h)
(0 + h)
sin h
=1
h→0 h
= lim
Since LHL 6= RHL, so limit does not exits.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Limit Principles
If limx→x0 f (x) = L1 and limx→x0 g(x) = L2 , then we have the following
1
For any constant c, limx→x0 c = c
2
limx→x0 [f (x)]n = [limx→x0 f (x)] = Ln1
3
limx→x0 [f (x)]1/n = [limx→x0 f (x)]
4
limx→x0 [f (x) ± g(x)] = limx→x0 f (x) ± limx→x0 g(x) = L1 ± L2
5
limx→x0 [f (x) × g(x)] = limx→x0 f (x) × limx→x0 g(x) = L1 × L2
h
i
limx→x0 f (x)
f (x)
L1
limx→x0 g(x)
= limx→x
g(x) = L2
6
n
1/n
1/n
= L1
here L1 ≥ 0
0
7
For a constant c, limx→x0 c × f (x) = c × limx→x0 f (x) = c × L1
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Limit Principles
If limx→x0 f (x) = L1 and limx→x0 g(x) = L2 , then we have the following
1
For any constant c, limx→x0 c = c
2
limx→x0 [f (x)]n = [limx→x0 f (x)] = Ln1
3
limx→x0 [f (x)]1/n = [limx→x0 f (x)]
4
limx→x0 [f (x) ± g(x)] = limx→x0 f (x) ± limx→x0 g(x) = L1 ± L2
5
limx→x0 [f (x) × g(x)] = limx→x0 f (x) × limx→x0 g(x) = L1 × L2
h
i
limx→x0 f (x)
f (x)
L1
limx→x0 g(x)
= limx→x
g(x) = L2
6
n
1/n
1/n
= L1
here L1 ≥ 0
0
7
For a constant c, limx→x0 c × f (x) = c × limx→x0 f (x) = c × L1
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Limit Principles
If limx→x0 f (x) = L1 and limx→x0 g(x) = L2 , then we have the following
1
For any constant c, limx→x0 c = c
2
limx→x0 [f (x)]n = [limx→x0 f (x)] = Ln1
3
limx→x0 [f (x)]1/n = [limx→x0 f (x)]
4
limx→x0 [f (x) ± g(x)] = limx→x0 f (x) ± limx→x0 g(x) = L1 ± L2
5
limx→x0 [f (x) × g(x)] = limx→x0 f (x) × limx→x0 g(x) = L1 × L2
h
i
limx→x0 f (x)
f (x)
L1
limx→x0 g(x)
= limx→x
g(x) = L2
6
n
1/n
1/n
= L1
here L1 ≥ 0
0
7
For a constant c, limx→x0 c × f (x) = c × limx→x0 f (x) = c × L1
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Limit Principles
If limx→x0 f (x) = L1 and limx→x0 g(x) = L2 , then we have the following
1
For any constant c, limx→x0 c = c
2
limx→x0 [f (x)]n = [limx→x0 f (x)] = Ln1
3
limx→x0 [f (x)]1/n = [limx→x0 f (x)]
4
limx→x0 [f (x) ± g(x)] = limx→x0 f (x) ± limx→x0 g(x) = L1 ± L2
5
limx→x0 [f (x) × g(x)] = limx→x0 f (x) × limx→x0 g(x) = L1 × L2
h
i
limx→x0 f (x)
f (x)
L1
limx→x0 g(x)
= limx→x
g(x) = L2
6
n
1/n
1/n
= L1
here L1 ≥ 0
0
7
For a constant c, limx→x0 c × f (x) = c × limx→x0 f (x) = c × L1
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Limit Principles
If limx→x0 f (x) = L1 and limx→x0 g(x) = L2 , then we have the following
1
For any constant c, limx→x0 c = c
2
limx→x0 [f (x)]n = [limx→x0 f (x)] = Ln1
3
limx→x0 [f (x)]1/n = [limx→x0 f (x)]
4
limx→x0 [f (x) ± g(x)] = limx→x0 f (x) ± limx→x0 g(x) = L1 ± L2
5
limx→x0 [f (x) × g(x)] = limx→x0 f (x) × limx→x0 g(x) = L1 × L2
h
i
limx→x0 f (x)
f (x)
L1
limx→x0 g(x)
= limx→x
g(x) = L2
6
n
1/n
1/n
= L1
here L1 ≥ 0
0
7
For a constant c, limx→x0 c × f (x) = c × limx→x0 f (x) = c × L1
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Limit Principles
If limx→x0 f (x) = L1 and limx→x0 g(x) = L2 , then we have the following
1
For any constant c, limx→x0 c = c
2
limx→x0 [f (x)]n = [limx→x0 f (x)] = Ln1
3
limx→x0 [f (x)]1/n = [limx→x0 f (x)]
4
limx→x0 [f (x) ± g(x)] = limx→x0 f (x) ± limx→x0 g(x) = L1 ± L2
5
limx→x0 [f (x) × g(x)] = limx→x0 f (x) × limx→x0 g(x) = L1 × L2
h
i
limx→x0 f (x)
f (x)
L1
limx→x0 g(x)
= limx→x
g(x) = L2
6
n
1/n
1/n
= L1
here L1 ≥ 0
0
7
For a constant c, limx→x0 c × f (x) = c × limx→x0 f (x) = c × L1
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Limit Principles
If limx→x0 f (x) = L1 and limx→x0 g(x) = L2 , then we have the following
1
For any constant c, limx→x0 c = c
2
limx→x0 [f (x)]n = [limx→x0 f (x)] = Ln1
3
limx→x0 [f (x)]1/n = [limx→x0 f (x)]
4
limx→x0 [f (x) ± g(x)] = limx→x0 f (x) ± limx→x0 g(x) = L1 ± L2
5
limx→x0 [f (x) × g(x)] = limx→x0 f (x) × limx→x0 g(x) = L1 × L2
h
i
limx→x0 f (x)
f (x)
L1
limx→x0 g(x)
= limx→x
g(x) = L2
6
n
1/n
1/n
= L1
here L1 ≥ 0
0
7
For a constant c, limx→x0 c × f (x) = c × limx→x0 f (x) = c × L1
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
Limit Principles
If limx→x0 f (x) = L1 and limx→x0 g(x) = L2 , then we have the following
1
For any constant c, limx→x0 c = c
2
limx→x0 [f (x)]n = [limx→x0 f (x)] = Ln1
3
limx→x0 [f (x)]1/n = [limx→x0 f (x)]
4
limx→x0 [f (x) ± g(x)] = limx→x0 f (x) ± limx→x0 g(x) = L1 ± L2
5
limx→x0 [f (x) × g(x)] = limx→x0 f (x) × limx→x0 g(x) = L1 × L2
h
i
limx→x0 f (x)
f (x)
L1
limx→x0 g(x)
= limx→x
g(x) = L2
6
n
1/n
1/n
= L1
here L1 ≥ 0
0
7
For a constant c, limx→x0 c × f (x) = c × limx→x0 f (x) = c × L1
Dr Mayank Goel
Essential Calculus
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find
√
limx→1
=
=
=
=
2x − 1
.
3x + 5
√
limx→1 2x − 1
limx→1 (3x + 5)
p
limx→1 (2x − 1)
limx→1 (3x + 5)
√
2 limx→1 x − limx→1 1
3 limx→1 x + limx→1 5
√
2×1−1
3×1+5
= 1/8
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find
√
limx→1
=
=
=
=
2x − 1
.
3x + 5
√
limx→1 2x − 1
limx→1 (3x + 5)
p
limx→1 (2x − 1)
limx→1 (3x + 5)
√
2 limx→1 x − limx→1 1
3 limx→1 x + limx→1 5
√
2×1−1
3×1+5
= 1/8
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find
√
limx→1
=
=
=
=
2x − 1
.
3x + 5
√
limx→1 2x − 1
limx→1 (3x + 5)
p
limx→1 (2x − 1)
limx→1 (3x + 5)
√
2 limx→1 x − limx→1 1
3 limx→1 x + limx→1 5
√
2×1−1
3×1+5
= 1/8
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find
√
limx→1
=
=
=
=
2x − 1
.
3x + 5
√
limx→1 2x − 1
limx→1 (3x + 5)
p
limx→1 (2x − 1)
limx→1 (3x + 5)
√
2 limx→1 x − limx→1 1
3 limx→1 x + limx→1 5
√
2×1−1
3×1+5
= 1/8
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find
√
limx→1
=
=
=
=
2x − 1
.
3x + 5
√
limx→1 2x − 1
limx→1 (3x + 5)
p
limx→1 (2x − 1)
limx→1 (3x + 5)
√
2 limx→1 x − limx→1 1
3 limx→1 x + limx→1 5
√
2×1−1
3×1+5
= 1/8
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find
√
limx→1
=
=
=
=
2x − 1
.
3x + 5
√
limx→1 2x − 1
limx→1 (3x + 5)
p
limx→1 (2x − 1)
limx→1 (3x + 5)
√
2 limx→1 x − limx→1 1
3 limx→1 x + limx→1 5
√
2×1−1
3×1+5
= 1/8
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find limx→0 f (x), where
(
0,
x =0
.
f (x) = |x|
x , elsewhere


x =0
0,
Function can be rewritten as f (x) = −1, x < 0 .


1,
x >0
LHL = limx→0− f (x) = limh→0 f (0 − h)
= limh→0 −1 = −1
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find limx→0 f (x), where
(
0,
x =0
.
f (x) = |x|
x , elsewhere


x =0
0,
Function can be rewritten as f (x) = −1, x < 0 .


1,
x >0
LHL = limx→0− f (x) = limh→0 f (0 − h)
= limh→0 −1 = −1
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Examples
Example
Find limx→0 f (x), where
(
0,
x =0
.
f (x) = |x|
x , elsewhere


x =0
0,
Function can be rewritten as f (x) = −1, x < 0 .


1,
x >0
LHL = limx→0− f (x) = limh→0 f (0 − h)
= limh→0 −1 = −1
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
RHL = limx→0+ f (x) = limh→0 f (0 + h)
= limh→0 1 = 1
Limit does not exists.
Observation
Limit of the function does not
depends upon the value of the
function at that point.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
RHL = limx→0+ f (x) = limh→0 f (0 + h)
= limh→0 1 = 1
Limit does not exists.
Observation
Limit of the function does not
depends upon the value of the
function at that point.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Examples
RHL = limx→0+ f (x) = limh→0 f (0 + h)
= limh→0 1 = 1
Limit does not exists.
Observation
Limit of the function does not
depends upon the value of the
function at that point.
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Find the limit
1
limx→0
1
x
Answer: LHL = −∞, RHL = ∞, so limit does not exists
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Find the limit
1
limx→0
1
x
Answer: LHL = −∞, RHL = ∞, so limit does not exists
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Find the limit
2
limx→0
1
x2
Answer: LHL = ∞, RHL = ∞, so limit does not exists
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Limit Principles
Find the limit
2
limx→0
1
x2
Answer: LHL = ∞, RHL = ∞, so limit does not exists
Dr Mayank Goel
Essential Calculus
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Find the limit
3
limx→0
| sin x|
sin x
Answer: LHL = −1, RHL = 1, limit does not exists
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Find the limit
3
limx→0
| sin x|
sin x
Answer: LHL = −1, RHL = 1, limit does not exists
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Find the limit
4
limx→0 sin πx
Answer: limit does not exists
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
Find the limit
4
limx→0 sin πx
Answer: limit does not exists
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples
Introduction to Limits
Limit Definition
Evaluation of Limits
THANK YOU
Dr Mayank Goel
Essential Calculus
Limit Principles
Examples