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NCERT Class 11 Maths Chapter 12 Notes: Limits and Derivatives

Limits and Derivatives are fundamental concepts in calculus. A limit helps us understand the behaviour of a function as the input approaches a particular value, while a derivative measures how a function changes; in other words, it's the rate of change or slope of a curve at a point.

Limits

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

We say $\lim\limits _{x \rightarrow a^{-}} f(x)$ is the expected value of $f$ at $x=a$ given the values of $f$ near to the left of $a$. This value is called the left-hand limit of $f$ at $a$.

We say $\lim\limits _{x \rightarrow a^{+}} f(x)$ is the expected value of $f$ at $x=a$ given the values of $f$ near to the right of $a$. This value is called the right-hand limit of $f$ at $a$.
If the right and left-hand limits coincide, we call the common value the limit of $f$ at $x=a$ and denote it by $\lim\limits _{x \rightarrow a} f(x)$.

If y=f(x) is a function where x=a, then it cannot take exact values, but closure to a, then limits are used to obtain a unique number when x tends to a or some value that is definite.

Mathematical representation: $\lim\limits _{x \rightarrow a} f(x)$

Left-Hand Limit:

If a function has values close to x=a on the left-hand, then the unique value is given by:

$f\left(a^{-}\right)=\lim\limits _{x \rightarrow a^{-}} f(x)=\lim\limits _{h \rightarrow 0} f(a-h)$

Right-Hand Limit:

If a function that is given has values close to x=a on the right-hand, then the unique value is given by:

$f\left(a^{+}\right)=\lim\limits _{x \rightarrow a^{+}} f(x)=\lim\limits _{h \rightarrow 0} f(a+h)$

Algebra of Limits $\lim\limits _{x \rightarrow a} f(x)$

Theorem 1:

If f and g are two functions where both limits $\lim\limits _{x \rightarrow a} f(x)$ and $\lim\limits _{x \rightarrow a} g(x)$ exist.

i) The limit of the sum of two functions is equal to the limit of the individual sums of each function.

$\lim\limits _{x \rightarrow a}[f(x)+g(x)] = \lim\limits _{x \rightarrow a} f(x) + \lim\limits _{x \rightarrow a} g(x)$

ii) The limit of the difference between two functions is equal to the limit of the individual differences of each function.

$\lim\limits _{x \rightarrow a}[f(x)-g(x)] = \lim\limits _{x \rightarrow a} f(x) - \lim\limits _{x \rightarrow a} g(x)$

iii) The limit of the product of two functions is equal to the limit of the individual product of each function.

$\lim\limits _{x \rightarrow a}[f(x).g(x)] = \lim\limits _{x \rightarrow a} f(x) . \lim\limits _{x \rightarrow a} g(x)$

iv) The limit of the quotient of two functions is equal to the quotient of the limit of each function.

$\lim\limits _{x \rightarrow a}[\frac{f(x)}{g(x)}] = \frac{\lim\limits _{x \rightarrow a} f(x)}{\lim\limits _{x \rightarrow a} g(x)}$ [Provided $\lim\limits _{x \rightarrow a} g(x)\neq0$]

v) $\lim\limits _{x \rightarrow a} (λ.f(x))=λ.\lim\limits _{x \rightarrow a} f(x)$

Limits of polynomials and rational functions

If $f$ is a polynomial function, then $\lim\limits _{x \rightarrow a} f(x)$ exists and is given by

$\lim\limits _{x \rightarrow a} f(x)=f(a)$

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

$\lim\limits _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n a^{n-1}$

Remark: The above expression remains valid for any rational number provided that ' $a$ ' 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) $\lim\limits _{x \rightarrow 0} \frac{\sin x}{x}=1$
(ii) $\lim\limits _{x \rightarrow 0} \cos x=1$
(iii) $\lim\limits _{x \rightarrow 0} \sin x=0$

Some Standard Limit Functions to Remember:

1. lim_x→a [(xⁿ-aⁿ)/(x-a)]= n.a^n-1

2. lim_x→0 [(sin x)/x] = 1

3. lim_x→0 [(tan x)/x] = 1

4. lim_x→0 [e^x/x] = 1

5. lim_x→0 [a^x-1)/x] = log_ea

6. lim_x→0 [(log(1+x))/x] = 1

Limits of Polynomial and Rational Functions:

1. lim_x→a x = a

2. lim_x→a xⁿ=aⁿ

3. When f(x) = g(x)/h(x) then

lim_x→a f(x) = lim_x→a [g(x)/h(x)] = [lim_x→a g(x)] / [lim_x→a h(x)] =g(a)/h(a)

Limits of Trigonometric Functions:

Theorem 2:

If f(x) and g(x) are two real-valued functions with the same domain satisfying

f(x)≤g(x) and limit of two functions lim_x→a f(x) and lim_x→a g(x) exists then

lim_x→a f(x) ≤ lim_x→a g(x)

Theorem 3: (Sandwich Theorem)

If f(x), g(x), hand (x) are real-valued functions with the same domain satisfying

f(x)≤g(x)≤h(x), if lim_x→a f(x) = l = lim_x→a h(x) then lim_x→a g(x) =l

Derivatives

1. If f is a real-valued function and a is the domain, then the derivative of f

lim_h→0 (f(a+h)-f(a))/h exists.

The derivative of f(x) is denoted by f’(a).

Therefore : f’(a) = lim_h→0 (f(a+h)-f(a))/h

2. If f is a real-valued function, then limits exist for

f(x) = lim_h→0 (f(x+h)-f(x))/h

Algebra of Derivatives of Functions

1. The derivative of the sum of two functions is equal to the sum of the individual derivatives of each function.

d/dx [f(x)+g(x)] = (d/dx) f(x) + (d/dx) g(x)

2. The derivative of the difference of two functions is equal to the derivative of the individual difference of each function.

(d/dx) [f(x)-g(x)] = (d/dx) f(x) - (d/dx) g(x)

3. The derivative of the product of two functions is equal to the derivative of the individual product of each function.

(d/dx) [f(x).g(x)] = (d/dx) f(x) . g(x)+f(x). (d/dx) g(x)

4. The derivative of the quotient of two functions is equal to the quotient of the derivative of each function.

(d/dx) [f(x)/g(x)] = [(d/dx) f(x) . g(x)-f(x). (d/dx) g(x)] / (g(x))²

Derivative of Polynomial and Trigonometric Functions:

If f(x) is said to be a polynomial function such that

f(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ……………….+ a₁x+a₀

Then the derivative of the function is given by:

(d/dx) (f(x)= na_nx^n-1+ (n-1)a_n-1x^n-2+(n-2)a_n-2x^n-3+..............+2a₂x+a₁

Some standard Derivatives:

1. $(d / d x)\left(x^n\right)=n x^{n-1}$
2. $(d / d x)(\sin x)=\cos x$
3. $(d / d x)(\cos x)=-\sin x$
4. $(d / d x)(\tan x)=\sec ^2 x$
5. $(d / d x)(\cot x)=-\operatorname{cosec}^2 x$
6. $(d / d x)(\sec x)=\sec x \cdot \tan x$
7. $(d / d x)(\sec x)=-\operatorname{cosec} x \cdot \cot x$
8. $(d / d x)\left(a^x\right)=a^x \log _{e^a}$
9. $(d / d x)\left(e^x\right)=e^x$
10. $(d / d x)\left(\log _e x\right)=1 / x$

With this topic, we conclude the NCERT Class 11 maths chapter 12 notes.

Limits and Derivatives: Previous Year Question and Answer

Given below are some previous year question answers of various examinations from the NCERT class 11 chapter 12, Limits and Derivatives:

Question 1: If $f(x)=1-x+x^2-x^3 \ldots-x^{99}+x^{100}$, then $\mathrm{f}^{\prime}(1)$ is equal to

Solution:
Given,
$\begin{aligned} & f(x)=1-x+x^2-x^3+\ldots-x^{99}+x^{100} \\ & f^{\prime}(x)=0-1+2 x-3 x^2+\ldots-99 x^{98}+100 x^{99} \\ & f^{\prime}(1)=-1+2-3+4-\ldots-99+100 \\ & =(2-1)+(4-3)+(6-5)+\ldots(100-99)=1+1+\ldots+1 \quad(50 \text { times })=50\end{aligned}$

Hence, the correct answer is 50.

Question 2: If $y=\frac{\sin (x+9)}{\cos x}$ then $\frac{d y}{d x}$ at $x=0$ is

Solution:
$\begin{aligned} & y=\frac{\sin (x+9)}{\cos x} \\ & \frac{d y}{d x}=\frac{\cos x \cos (x+9)-\sin (x+9)(-\sin x)}{\cos ^2 x} \\ & =\frac{\cos x \cos (x+9)+\sin (x+9) \sin x}{\cos ^2 x} \\ & =\frac{\cos (x+9-x)}{\cos ^2 x}=\frac{\cos 9^{\circ}}{\cos ^2 x} \\ & \left(\frac{d y}{d x}\right)_{x=0}=\frac{\cos 9^{\circ}}{(1)^2}=\cos 9^{\circ}\end{aligned}$

Hence, the correct answer is $\cos 9^{\circ}$.

Question 3: If $f(x)=1+x+\frac{x^2}{2}+\frac{x^3}{3}+\ldots+\frac{x^{100}}{100}$ then $\mathrm{f}^{\prime}(1)$ is equal to:

Solution:
$\begin{aligned} & f(x)=1+x+\frac{x^2}{2}+\frac{x^3}{3}+\ldots+\frac{x^{100}}{100} \\ & f^{\prime}(x)=0+1+\frac{2 x}{2}+\frac{3 x^2}{3}+\ldots+\frac{100 x^{99}}{100} \\ & f^{\prime}(x)=0+1+x+x^2+\ldots x^{99} \\ & f^{\prime}(1)=1+1+1+\ldots+1 \quad(100 \text { times })=100\end{aligned}$

Hence, the correct answer is 100.

Importance of NCERT Class 11 Maths Chapter 12 Notes:

NCERT Class 11 Maths Chapter 12 notes will be very helpful for students to score good marks in their 11th class exams and help in building a strong base for competitive exams. Also, it is the basis of differentiation to be studied in class 12th.

The CBSE Class 11 Maths Chapter 12 will help to understand the formulas, statements, and rules with their conditions in detail. You can get compact knowledge as a document in the Limits and Derivatives Class 11 chapter 12 PDF download.

NCERT problems are very important in order to perform well in exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives. Students are advised to go through the NCERT Class 11 Maths Chapter 12 Notes before solving the questions.

These NCERT notes are very useful for boosting your exam preparation and quick revision.

NCERT Class 11 Notes Chapter Wise

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Subject-Wise NCERT Exemplar Solutions

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