CHAPTER # 01
Limits
Syllabus
Part 2
Part 1
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Limit
Continuity
Differentiation
Higher order derivatives
Integration
Partial differentiation
Ordinary differential equations
Algebra
• Gauss Elimination
method
• Gauss Jordan method
• Cramer rule
• Eigen values problem
Concept of Limits

It is often useful to find function values of a function f(X) when x is close to a number a, but not
necessarily equal to a.

Limits are all about determining how a function behaves as it approaches a specific point or
value
Let consider
with a = 2. Note that 2 is not in the domain of I, since substituting x = 2
gives us the undefined expression 0/0.Let consider the following tables,
list some function values (to eight-decimal-place accuracy)
for x close to 2.
Concept of limits
Notice that as
the values of
x approach 2 fr
om either side
of 2, the values
of f(X)
approaches 4/3
It appears that the closer x is to 2, the closer I(x) is to 4/3
Using this limit notation , we may denote the result of our illustration as
follows:
Mathematical
representation
Limits concept
Notation
Intuitative meaning
Graphical illustrations
We can make f(X) as close
to L as desired by choosing
X close to a as possible but
X not equal to a
If fIx) approaches some number as x approaches a. But we do not know what
that number is, we use the phrase limx _ a f(x) exisls.
Limits concept
Note that substituting 0 for x gives us the undefined expression 0/0.
Let find different functional values of f(X) from calculator
The table
indicates
that (sin xl/x
gets closer to I
as x gets
closer to 0
Limits

A limit of a function is a number that a function reaches as the
independent variable of the function reaches a given value. The value (say
a) to which the function f(x) gets close arbitrarily as the value of the
independent variable x becomes close arbitrarily to a given value
“A” symbolized as f(x) = A.
You can say
limit of this
function is 32
as X
approaches
to 2
Limits
We write it as
The sentence limx→c fx L is read, “The limit of f of x as x
approaches c equals L.”
Limits

The limit value of a function does not depend on how the function is defined at the point
being approached
The function ƒ has limit
2 as x approaches 1
even though ƒ is not
defined at x = 1.
The function g has
limit 2 as x approachd1
even though 2 ≠ g(1).
The function h is the only
one of the three
functions in whose limit as
x approaches 1 equals its
value at x = 1
The limit laws
Example
The number 2 is not in the domain off since the meaningless expression 0/0 is obtained if
2 is substituted for x. Factoring the numerator and denominator gives us and taking limit
as
One sided limits
Left hand limit
Right hand limit
Limit Theorem
Continued
One sided limits are
Since the right-hand and left-hand limits are equal, it follows from
Theorem that
Note that the
function value
f(1) = 4 is
irrelevant in
finding the
limit.
Examples
Find the limit of
Evaluate
Sandwich theorem
Suppose that g(x) < ƒ(x)< h(x) for all x in some open interval containing c, except possibly at x = c itself.
Suppose also that
The Sandwich
Theorem is
also called the
Squeeze
Theorem or
the Pinching
Theorem.
Example
Try these problems