NCERT Class 11-Math՚s: Chapter – 13 Limits and Derivatives Part 12 (For CBSE, ICSE, IAS, NET, NRA 2022)

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13.1 Overview

13.1. 1 Limits of a function

Let f be a function defined in a domain which we take to be an interval, say, I. We shall study the concept of limit of at a point ‘a’ in I.

We say is the expected value of at given the values of near to the left of a. This value is called the left hand limit of at a.

We say is the expected value of at given the values of f near to the right of . This value is called the right hand limit of at .

If the right and left hand limits coincide, we call the common value as the limit of at and denote it by

Let and be two functions such that both exist. Then

(iii) For every real number

Limits of polynomials and rational functions

If is a polynomial function, then exists and is given by

An Important limit

An important limit which is very useful and used in the sequel is given below:

Remark The above expression remains valid for any rational number provided ‘a’ is positive.

Limits of trigonometric functions

To evaluate the limits of trigonometric functions, we shall make use of the following limits which are given below:

(i) (ii) (iii)

13.1. 2 Derivatives

Suppose f is a real valued function, then

is called the derivative of f at x, provided the limit on the R. H. S. of (1) exists. Algebra of derivative of functions Since the very definition of derivatives involve limits in a rather direct fashion, we expect the rules of derivatives to follow closely that of limits as given below:

Let f and g be two functions such that their derivatives are defined in a common domain. Then:

(i) Derivative of the sum of two functions is the sum of the derivatives of the functions.

(ii) Derivative of the difference of two functions is the difference of the derivatives of the functions.

(ii) Derivative of the product of two functions is given by the following product rule.

This is referred to as Leibnitz Rule for the product of two functions.

(iv) Derivative of quotient of two functions is given by the following quotient rule (wherever the denominator is non-zero) .