Limits in Maths presentation

PRESENTED TO:
SIR NASIR NADEEM

PRESENTED BY:
AMNA LATIF HUSS19112019 [BS-PHED]
ANAMTA MEHBOOB HUSS 19112020 [BS-PHED]
SABA SHAHBAZ HUSS19112021 [BS-PHED]
ADEENA MAHAM HUSS19112023 [BS-PHED]
KINZA SHAHBAZ HUSS19112025 [BS-PHED]
MADIHA KANWAL HUSS191120 [BS-PHED]
SHEEZA KOMAL HUSS19112029 [BS-PHED]
KHALDA BIBI HUSS19112031 [BS-PHED]
AYESHA JAMEEL HUSS19112032 [BS-PHED]
AIMA KHIZER HUSS19114004 [BS-ISL]
QUDSIA KANWAL HUSS 19114006 [BS-ISL]

TODAY WE PRESENTS
DEFINE LIMITS
PROPERTIES OF LIMIT
ONE SIDE LIMIT

DEFINE LIMITS
• If f (x) is function of x and c, L are the real
number, then L is the limit of a function f (x) as x
approaches c:
lim 𝑥→𝑐 𝑓 (x) = L

A limit is the value that a function (or sequence)
"approaches" as the input "approaches" some
value. Limits are essential
to calculus (and mathematical analysis in
general) and are used to define continuity,
derivatives, and egralsint.

EXAMPLE
• lim𝑥→−2 ( 𝑋 2 + 3x − 7 ) Solution:
Apply the limits = ( − 2 ) 2 + 3( -2) − 7
= 4 − 6 − 7
= 4 − 13
= − 9 Ans

What is precise definition of limit?
Lets start by starting that f(x) is a function
on an open interval that contains x=ax=a
but that the function dose not necessarily
exist at x=ax=a.

PROPERTIES AND LAW OF LIMITS
• SUM RULE
• DEFRENCE RULE
• PRODUCT RULE
• QUOTIENT RULE
• CONSTANT RULE

SUM RULE
• This rule states that the limit of the sum of two functions is equal to the
sum of their limits:

DEFRENCE RULE
• The limit of a difference is equal to the difference of the limits.

PRODUCT RULE
• This rule says that the limit of the product of two functions is the product
of their limits (if they exist)

QUOTIENT RULE
• The limit of quotient of two functions is the quotient of their limits,
provided that the limit in the denominator function is not zero

CONSTANT RULE
• The limit of a constant function is the constant:

One-sided limit
A one-sided limit is either of the two limits of a
function f(x) of a real variable x as x approaches a
specified point either from the left or from the right.
does not exist, the two one-sided limits nonetheless
exist. Consequently, the limit as x approaches a is
sometimes called a "two-sided limit".

TYPES OF LIMITS
There are two types or limits
• Right Hand Limit
• Left Hand Limit

RIGHT HAND LIMIT
• If x approach to “ a “ through value of x greater
then “ a “ we say that x approach through the
right and written as:
X → a+0 or X → 𝑎 +

LEFT HAND LIMIT
• If x approach to “ a “ through value of x less then “
a “ we say that x approach through the left and
written as:
X → a − 0 or X → 𝑎 −

• Left Hand Limit:
lim 𝑥→𝑎− 𝑓 (𝑥)
• Right Hand Limit :
lim 𝑥→𝑎+ 𝑓 (𝑥)
• If L.H.L = R.H.L
Then lim 𝑥→𝑎 𝑓 (𝑥) exist

EXAMPLE
• f (x) = |x|/x at x = 1
lim 𝑥→1− | 𝑋 | 𝑋 = -1
lim 𝑥→1+ | 𝑋 | 𝑋 = 1
The left and right limits are different, therefore limit does not
exist..

TECHNIQES OF CALCULATING LIMITS
• DIRECT SUBSTITUTION
• FACTORIZING
• RETIONALIZATION
• INFINITY
• TRIGNOMETRIC LIMITS
• NUMBER [ e ]

DIRECT SUBSTITUTION
One of the most easiest and useful ways to evaluate a
limit analytically is substitution. Direct substitution is a
valid method to evaluate limit.
If p is polynomial and c is a real number.
If r is a rational number.
For example

RETIONALIZATION
Rationalization generally means to multiply
A rational function by a clever form of one in
order to eliminate radical symbols or imaginary
numbers in the denominator. It is also used to
evaluate limits in order to avoid having a zero in
the denominator when you substitute.

Limits at Infinity.
Limits at infinity are used to describe the
behavior of functions as the independent variable
increases or decreases without bound. If a
function approaches a numerical value L in either
of these situations, write. And f(x) is said to have a
horizontal asymptote at y = L.

Limits in Maths presentation

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Limits in Maths presentation