TEACHER ORIENTATION
TOPIC:LIMITS
OBJECTIVE- To understand the new concept of the
mathematics which deals with the study of change
in value of a function as the point in domain
change. By learning the concepts of the new topic
Limits and Derivatives student will able to use its
knowledge in higher classes studies in engineering
and other branches like economic and statistics
INTRODUCTION: It is the part of mathematics which
mainly deals with the study of change in the value of a
function as the value of the variable in the domain
change. Calculus has a very wide range of uses in
sciences, engineering, economics and many other walks
of life.
In this chapter we shall introduce the concept of
limit of a real function, study some algebra of limits and
will evaluate limits of some algebraic and trigonometric
functions, then we shall define the derivative of a real
function give its geometrical and physical interpretation,
study some algebra of derivatives and will obtain
derivatives of some algebraic and trigonometric
functions.
The concept of limit originated in course of
developing a general technique to find the tangent to a
curve and also in course of finding the velocity of a
moving particle at a particular instant when the distance
of the particle changes with time. But here we shall first
discuss the concept of limit and then the concept of
derivative. The concept of limit is an abstract one. We
will explain the intuitive concept of limit with the help of
some examples. For this we shall first define some terms
and then consider some examples.
MEANING OF TENDS TO:
1. x
a – 0 or x
a – : This is read as x tends to a from left.
This means that x Is very very close to a but it is always less than
a.
2. x
a + 0 or x
a + : This is read as x tends to a from
right. This means that x is very very close to a but it is always
greater than a.
3. x
a : This is read as x tends to a. this means that x is very
very close to a but it is never equal to a. here x tends to a either
from left or from right.
-∞
∞
x
a–0
-∞
∞
a+0
x
-∞
∞
x
a
x
CONCEPTS OF LIMIT:
1. We consider a function defined by f(x)= x2 – 4
x–2
Here f(x) is defined for all x except x=2
At x=2, f(x) =
22 −4
2−2
=
0
0
Thus at x=2, f(x) is not defined
When x≠2
Then f(x) =
𝑥 2 −4
𝑥−2
=
(𝑥+2)(𝑥−2)
𝑥−2
= x+2
We consider the following tables-1 and table-2
When x≠2 and x→2− and x→2+
Table-1
X
1.9
1.99
1.999
1.9999
F(x)=x+2 3.9
3.99
3.999
3.9999
Table-2
X
2.1
2.01
2.001
2.0001
F(x)=x+2 4.1
4.01
4.001
4.0001
1.99999
3.99999
2.00001
4.00001
It is clear from table-1 from the graph of the function f
that f(x) →4 for all x→2− ,we express this fact by saying
that lim− 𝑓(𝑥) = 4
𝑥→2
Similarly from table -2 , We have lim⁡x 2+ f(x) =4
→
In fact, as x→ 2 either from left or from right , f(x)→ 4
as shown in fig. by two arrow heads , we express this
fact by saying that lim 𝑓 (𝑥 ) = 4
𝑥→2
2.
Consider the function f defined by f(x) = x2
Clearly, domain of f = R, so that the function value f(x)
can be obtained for every value of x.
Let us investigate the function values f(x) when x is near
to 0. Let x take on values nearer and nearer to 0 either
from left side or from right side which we illustrate by
means of tables given below:
Table 3X
f(x)
-0.5
0.25
-0.1
0.01
-0.01
-0.001
0.0001 0.000001
…
…
Table 4X
f(x)
0.5
0.25
0.1
0.01
0.01
0.001
0.0001 0.000001
…
…
A portion of the graph of the function f is shown in the
figure:
It is clear from the table 3 or from the graph of the
function f that f(x) is arbitrarily near the value for all x
sufficiently near the number 0 (from left). We express
this fact by saying that
lim 𝑓(𝑥 ) = 0
𝑥→0−
Also, it is clear from the table 4 or from the graph of the
function f that f(x) is arbitrarily near the value 0 for all x
sufficiently near the number 0 (from right). We express
this fact by saying that
lim 𝑓(𝑥 ) = 0
𝑥→0+
In fact, as x⁡→ 0 either from left or from right, f(x) →⁡0, as
shown in the figure sy two arrow heads. We express this
fact by saying that lim 𝑓 (𝑥 ) = 0.
𝑥→0
3. Consider the following function f defined by
𝑥 − 2,
𝑥<0
0,
𝑥=0
𝑓(𝑥 ) = ⁡ {
𝑥 + 2, 𝑥 > 0
Table 5x
-0.5
f(x)
-2.5
-0.1
-0.01
-2.1
-2.01
-0.001
0.0001
-2.001
2.0001
…
…
It is clear from the table 5 that f(x) is arbitrarily near the
value -2 for all x sufficiently near the number 0 (from
left). We express this fact by saying that lim 𝑓 (𝑥 ) = −2
𝑥→0−
Now consider the following table:
Table 6x
0.5
f(x)
2.5
0.1
2.1
0.01
2.01
0.001 0.0001
2.001 2.0001
…
…
It is clear from the table 5 that f(x) is arbitrarily near the
value 2 for all x sufficiently near the number 0 (from
right). We express this fact by saying that lim 𝑓(𝑥 ) = 2
𝑥→0+
This example shows that lim 𝑓 (𝑥 ) ≠ lim 𝑓 (𝑥 ), which
𝑥→0−
𝑥→0+
we express by saying that lim 𝑓(𝑥 ) does not exist.
𝑥→0
4. let
then f(1) is not defined (see division by zero), yet as x moves
arbitrarily close to 1, f(x) correspondingly approaches 2:
f(0.9) f(0.99) f(0.999) f(1.0)
1.900 1.990 1.999
f(1.001) f(1.01) f(1.1)
⇒ undefined ⇐ 2.001
2.010 2.100
Thus, f(x) can be made arbitrarily close to the limit of 2 just
by making x sufficiently close to 1.
In other words,
This can also be calculated algebraically,
for all real numbers x ≠ 1.
as
Now since x + 1 is continuous in x at 1, we can now plug in 1
for x, thus
.
In addition to limits at finite values, functions can also have
limits at infinity. For example, consider



f(100) = 1.9900
f(1000) = 1.9990
f(10000) = 1.99990
As x becomes extremely large, the value
of f(x) approaches 2, and the value of f(x) can be made as
close to 2 as one could wish just by picking x sufficiently
large. In this case, the limit of f(x) as x approaches infinity
is 2. In mathematical notation,
Hence, a function f is said to tend to a limit l as x
approaches c if the difference between f(x) and l can be
made as small as we please by taking x sufficiently near c
and we write it as lim 𝑓 (𝑥 ) = 𝑙.
𝑥→𝑐
Theorem: lim 𝑓 (𝑥 ) = 𝑙 iff lim 𝑓 (𝑥 ) = 𝑙 = lim 𝑓(𝑐 ) .
𝑥→𝑐
𝑥→𝑐+
𝑥→𝑐−
We assert that for the existence of lim 𝑓(𝑥 ), it is
𝑥→𝑐
essential that lim 𝑓 (𝑥 )⁡𝑎𝑛𝑑⁡ lim 𝑓 (𝑥 ) must both exist
𝑥→𝑐+
𝑥→𝑐−
separately and be equal. This equal value is the limit of
the function.
SOME STANDARD RESULTS ON LIMITS:
1. (i) lim 𝛼 = 𝛼, where 𝛼 is a fixed real number.
𝑋→𝐶
(ii) lim 𝑥 𝑛 = 𝑐 𝑛 , for all n ∈ N.
𝑥→𝑐
(iii) lim 𝑓 (𝑥 ) = 𝑓(𝑐), where f(x) is a real polynomial
𝑥→𝑐
in x.
2. ALGEBRA OF LIMITS
Let f, g be two functions such that lim 𝑓 (𝑥 ) = 𝑙 and
𝑥→𝑐
lim 𝑔(𝑥 ) = 𝑚, then
𝑥→𝑐
(i) lim (𝛼𝑓(𝑥 )) = 𝛼. lim 𝑓 (𝑥 ) = 𝛼𝑙, for all 𝛼⁡ ∈ 𝑅.
𝑥→𝑐
𝑥→𝑐
(ii) lim (𝑓(𝑥 ) + 𝑔(𝑥 )) = lim 𝑓 (𝑥 ) + lim 𝑔(𝑥 ) =
𝑥→𝑐
𝑥→𝑐
𝑥→𝑐
𝑙 + 𝑚.
(iii) lim (𝑓(𝑥 ) − 𝑔(𝑥 )) = lim 𝑓 (𝑥 ) − lim 𝑔(𝑥 ) =
𝑥→𝑐
𝑥→𝑐
𝑥→𝑐
𝑙−𝑚
(iv) lim (𝑓(𝑥 )𝑔(𝑥 )) = lim 𝑓(𝑥 ). lim 𝑔(𝑥 ) = 𝑙𝑚
𝑥→𝑐
(v) lim
𝑓(𝑥)
lim 𝑓(𝑥)
=𝑥→𝑐
𝑥→𝑐
𝑙
𝑥→𝑐
= ,provided m≠0
lim 𝑔(𝑥) 𝑚
𝑥→𝑐 𝑔(𝑥) 𝑥→𝑐
SANDWICH THEOREM—
If f,g@h are functions such that f(x)≤g(x)≤h(x) for all
x in some neighbourhood of c (except possibly at
x=c) and if lim 𝑓(𝑥 ) = 𝑙 =
𝑥→𝑐
lim ℎ(𝑥 ), 𝑡ℎ𝑒𝑛 lim 𝑔(𝑥 ) = 𝑙
𝑥→𝑐
𝑥→𝑐
Fig
Some Theorems on limits1. lim 𝑓 (𝑥 ) = 𝑐⁡⁡⁡
𝑥→𝑎
2. If f(x) =𝑎0 +𝑎1 x +𝑎2 𝑥 2 +𝑎3 𝑥 3 +…+𝑎𝑛 𝑥 𝑛 be a
polynomial in x ,then lim 𝑓(𝑥)= 𝑎0 +𝑎1 a +𝑎2 a +𝑎3 a
𝑥→𝑎
𝑛
+…+𝑎𝑛 𝑎 = f(a)
3. lim
𝑥 𝑛 −𝑎𝑛
𝑥→𝑎 𝑥−𝑎
=n𝑎𝑛−1 where n is a rational number
4. lim 𝑐𝑜𝑠𝜃= 1
𝜃→0
5. lim
𝑠𝑖𝑛𝜃
𝜃→0 𝜃
𝑡𝑎𝑛𝜃
6. lim
𝜃→0
𝜃
= 1 where 𝜃 is in radian
=1 where 𝜃 is in radian
7. lim 𝑠𝑖𝑛𝜃 = 0
𝜃→0
8. lim 𝑡𝑎𝑛𝜃 = 0
𝜃→0
9. lim
𝜃→0
1−𝑐𝑜𝑠𝜃
𝜃
=0
INSTANT DIAGONOSIS –
Evaluate the following limits:
1. lim (𝑥 2 + 𝑥)
𝑥→1
2. lim (3𝑥 3 − 5𝑥 + 2)
𝑥→2
3. lim
𝑥 3 −8
𝑥→2 𝑥−2
cos 𝑥
4. lim
𝑥→0 𝜋−𝑥
HOMEWORK:Evaluate the following limits:
1. lim 𝜋𝑟 2
𝑟→1
2. lim (𝑐𝑜𝑠𝑒𝑐⁡𝑥 − cot 𝑥)
𝑥→0
3. lim
𝑥→0
tan 𝑥−sin 𝑥
𝑥3
ASSIGNMENT –
Level 1
Q1. lim
𝑥 4 −16
𝑥→2 𝑥−2
√1+𝑥−1
Q2.lim
𝑥
𝑥→0
Level 2Q1. lim
𝑥 2 −4
𝑥→2 𝑥−2
cos 2𝑥−1
Q2. lim
𝑥→0 𝑐𝑜𝑠𝑥−1
Level 3𝑎𝑥+𝑥𝑐𝑜𝑠𝑥
Q1. lim
𝑥→0
𝑏𝑠𝑖𝑛𝑥
Q2. Find the derivatives of the following functions –
𝑥 2 𝑐𝑜𝑠
𝜋
4
𝑠𝑖𝑛𝑥
Q3. Find the derivatives of the following functions
from first principles –
sinx + cosx
PROJECT1. Give two functions f and g which are define as a limit
of the function at the given points.
2. Collect two algebraic functions f and g and verify
using the data of the value of the given point and by
drawing the graph the functions are defined at the
given point or not. If the limit of the function are
defined then find the limit of the function at the
given point.