# OP Malhotra Class-11 Limits S.Chand ISC Maths Solution

OP Malhotra Class-11 Limits S.Chand ISC Maths Solution Chapter-18. Step by step Solutions of OP Malhotra S.Chand ISC Class-11 Mathematics with Exe-18 (a), 18 (b), 18 (c), 18 (d), 18 (e), 18 (f), 18 (g), 18 (h), 18 (i), 18 (j), With Chapter Test. Visit official Website for detail information about ISC Board Class-12 Mathematics.

## OP Malhotra Class-11 Limits S.Chand ISC Maths Solution

 Class: 11th Subject: Mathematics Chapter  : Ch-18 Limits of Section -A
 Board ISC Writer OP Malhotra Publications S.Chand Publications 2020-21

: Select Topics :-

Exe-18 (a)

Exe-18 (b)

Exe-18 (c)

Exe-18 (d)

Exe-18 (e)

Exe-18 (f)

Exe-18 (g)

Exe-18 (h)

Exe-18 (i)

Exe-18 (j)

Chapter Test

### OP Malhotra Class-11 Limits S.Chand ISC Maths Solution

Limits Definition

A limit of a function f(x) is defined as a value, where the function reaches as the limit reaches some value. Limits are used to define integration, integral calculus and continuity of the function.

If f(y) is a function, then the limit of the function can be represented as;

limy→c

This is the general expression of limit, where c is any constant value. But there are some important properties of limits which we will discuss here.

• When we say that x approaches a (x → a) then we mean that the variable x takes those values which are either less than ‘a’ or greater than ‘a’ and the nu­merical difference between ‘x’ and ‘a’ can be made as small as we please

OR

Let y = f(x) be a function of x. If at x = a, f(x) takes indeterminate form, then we consider the values of the function which is very near to a. If these value tend to a definite unique number as x tends to a, then the unique number so obtained is called the limit of f(x) at x = a and we write it as lim𝑥𝑎𝑓(𝑥).

#### Limits

l is called the limit of the function f(x) if the equation is given as x → a, f(x) → l, and this is symbolically written for all the limits, the function should assume at a given point x = a. x could approach a number in two ways, either from the left or from the right, i.e., all the values of x near could be greater than or could be less than a.

Right-Hand Value

In this type of limits, Right-hand limit Value is referred to the situation in which f(x) gets dictated by values of f(x) when x tends to from the right.

#### What is a Left-Hand Value?

In the case of Left-hand limit, when tends to from the left, the value of f(x) gets dictated by values of f(x).

In this case, the right and left-hand limits are different, and hence we say that the limit of f(x) as x tends to zero does not exist (even though the function is defined at 0). This could also be followed in limits and continuity of values.

#### Derivatives

Derivatives are defined as the varying rate of change of a function with respect to an independent variable. The derivative is primarily used when there is some varying quantity, and the rate of change is not constant. The derivative is used to measure the sensitivity of one variable (dependent variable) with respect to another variable (independent variable).Derivatives Meaning Derivatives Maths refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount.First-Order Derivative The first order derivatives tell about the direction of the function whether the function is increasing or decreasing. The first derivative math or first-order derivative can be interpreted as an instantaneous rate of change. It can also be predicted from the slope of the tangent line.Second-Order Derivative The second-order derivatives are used to get an idea of the shape of the graph for the given function. The functions can be classified in terms of concavity. The concavity of the given graph function is classified into two types namely:

1. Concave Up
2. Concave Down.

Exe-18 (a)

Page 18-8

### Exe-18 (b)

OP Malhotra Class-11 Limits S.Chand ISC Maths Solution

Page 18-11

Exe-18 (c)

Page 18-13

### Exe-18 (d)

OP Malhotra Class-11 Limits S.Chand ISC Maths Solution

Page 18-15

Exe-18 (e)

Page 18-18

Exe-18 (f)

Page 18-22

### Exe-18 (g)

OP Malhotra Class-11 Limits S.Chand ISC Maths Solution

Page 18-27

Exe-18 (h)

Page 18-31

### Exe-18 (i)

OP Malhotra Class-11 Limits S.Chand ISC Maths Solution

Page 18-35 to 18-36

### Exe-18 (j)

Class-11 Limits S.Chand ISC Maths Solution

Page 18-38 to 18-39

Chapter Test

### OP Malhotra Class-11 Limits S.Chand ISC Maths Solution

Page 18-39

-: End of Limits Solution :-

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