# Limits And Derivatives class 11 Notes Mathematics

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## Limits And Derivatives class 11 Notes Mathematics

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## 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

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

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

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

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

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

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

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