# Limits And Derivatives class 11 Notes Mathematics

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## CBSE Guide Limits And Derivatives class 11 Maths Notes

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## Limits And Derivatives class 11 Notes Mathematics

Download CBSE class 11th revision notes for Chapter 13 Limits And Derivatives class 11 Notes Mathematics in PDF format for free. Download revision notes for Limits And Derivatives class 11 Notes Mathematics and score high in exams. These are the Limits And Derivatives class 11 Notes Mathematics prepared by team of expert teachers. The revision notes help you revise the whole chapter in minutes. Revising notes in exam days is on of the best tips recommended by teachers during exam days.

NCERT solutions for Class 11 Maths

## CBSE Class 11 Mathematics Revision Notes Chapter 13 Limits And Derivatives

1. Limits
2. Derivatives
3. Miscellaneous Questions

Meaning of $x \rightarrow a$ or “$x$ tends to $a$” or “$x$ approaches $a$“, $x$ is a variable. The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit. It can be changed so that its value comes nearer and nearer to $a$$0 < \left| {x - a} \right| < \delta$

(i) $x \ne a$,      (ii) $\left| {x - a} \right|$ becomes smaller and smaller as we please.

Neighbourhood: The set of all real numbers lying between $a - \delta$ and $a + \delta$ is called the neighbourhood of $a$. Neighbourhood of $a$ = $\left( {a - \delta ,a + \delta } \right),$     $x \in \left( {a - \delta ,a + \delta } \right)$

• Limit of a function at a point is the common value of the left and right hand limits, if they coincide

Left hand limit of $f$ at $x = a$. When $x$ approches $a$ from left hand side of $a$, the function $f(x)$ tends to $l$ “a definite number”. This definite number $l$ is said to be the left hand limit of $f$ at $x = a$.

Right hand limit of $f$ at $x = a$. When $x$ approches $a$ from right hand side of $a$, the function $f(x)$ tends to $l$ “a definite number”. This definite number $l$ is said to be the right hand limit of  at $x = a$.

Therefore, if Left hand limit of $f$ at $x = a$ = Right hand limit of $f$ at $x = a$, then the limit of $f(x)$ at $x = a$ exists.

• For function f and a real number a, $\underset{x\to \infty }{\mathop{\lim }}\,f(x)$ and f (a) may not be same (Infact, one may be defined and not the other one).
• For functions f and g the following holds:.

$\mathop {\lim }\limits_{x \to a} \left[ {f(x) \pm g(x)} \right] =$$\mathop {\lim }\limits_{x \to a} f(x) \pm \mathop {\lim }\limits_{x \to a} g(x)$

$\underset{x\to a}{\mathop{\lim }}\,\left[ f(x).g(x) \right]=$$\underset{x\to a}{\mathop{\lim }}\,f(x).\underset{x\to a}{\mathop{\lim }}\,g(x)$

$\underset{x\to a}{\mathop{\lim }}\,\left[ \frac{f(x)}{g(x)} \right]=$$\frac{\underset{x\to a}{\mathop{\lim }}\,f(x)}{\underset{x\to \infty }{\mathop{\lim }}\,g(x)}$

### Following are some of the standard limits

$\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$

$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{x}=1$, $\mathop {\lim }\limits_{x \to a} {{\sin \left( {x - a} \right)} \over {x - a}} = 1$

$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{x}=0$

$\mathop {\lim }\limits_{x \to 0} {{\tan x} \over x} = 1,$$\mathop {\lim }\limits_{x \to a} {{\tan \left( {x - a} \right)} \over {x - a}} = 1$

$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^{ - 1}}x} \over x} = 1,$$\mathop {\lim }\limits_{x \to 0} {{{{\tan }^{ - 1}}x} \over x} = 1$

$\mathop {\lim }\limits_{x \to 0} {{{a^x} - 1} \over x} = {\log _e}a,$$a > 0,a \ne 1$

### Derivatives

The derivative of a function f at a is defined by

$f'(a)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(a+h)-f(a)}{h}$

Derivative of a function f at any point x is defined by

$f'(x)=\frac{df(x)}{dx}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(x+h)-f(x)}{h}$

For functions u and v the following holds:

$(u\pm v)'=u'\pm v'$

$(uv)'=u'v+uv'$            $\Rightarrow$          ${d \over {dx}}\left( {uv} \right) = u.{{dv} \over {dx}} + v.{{du} \over {dx}}$

${{\left( \frac{u}{v} \right)}^{'}}=\frac{u'v-uv'}{{{v}^{2}}}$                 $\Rightarrow$          ${d \over {dx}}\left( {{u \over v}} \right) = {{v.{{du} \over {dx}} - u.{{dv} \over {dx}}} \over {{v^2}}}$

provided all are defind.

### Following are some of the standard derivatives

$\frac{d}{dx}({{x}^{n}})=n{{x}^{n-1}}$

$\frac{d}{dx}(\sin x)=\cos x$

$\frac{d}{dx}(\cos x)=-\sin x$

${d \over {dx}}\left( {\tan x} \right) = {\sec ^2}x$

${d \over {dx}}\left( {\cot x} \right) = - \cos e{c^2}x$

${d \over {dx}}\left( {\sec x} \right) = \sec x.\tan x$

${d \over {dx}}\left( {\cos ecx} \right) = - \cos ec{\rm{ }}x.\cot x$

## Limits And Derivatives class 11 Notes

• CBSE Revision notes for Class 11 Mathematics PDF
• Revision notes Class 11 Maths – CBSE
• CBSE Revisions notes and Key Points Class 11 Mathematics
• Summary of the NCERT books all chapters in Mathematics class 11
• Short notes for CBSE class 11th Mathematics
• Key notes and chapter summary of Mathematics class 11
• Quick revision notes for CBSE exams

## CBSE Class 11 Revision Notes and Key Points

Limits And Derivatives class 11 Notes Mathematics. CBSE quick revision note for class-11 Mathematics, Physics, Chemistry, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.

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