Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Limits and Continuity-cbse mathematics


Limits and Continuity 
Basic concepts and formulas based on limit and continuity important for the students of classes 11 and 12. Important formulas of chapter 13 class 11 and important formulas necessary for the students of class 12 chapter 5


Introduction

Limit and continuity is the introduction of calculus. Calculus is that branch of Mathematics which mainly deals with the study of change in the value of a function as the point in the domain changes.

 Definition: 

Suppose f is a real function on a subset of the real numbers and let c be a point in the domain of f. Then f is continuous at c if \[\lim_{x\rightarrow c}f(x)=f(c)\]

ALGEBRA OF LIMITS

\[Let \; f(x)\; and\; g(x)\; be\; two\; functions \; such\; that\; \lim_{x\rightarrow 0}\; f(x)\; and\; \lim_{x\rightarrow 0}\; g(x) \; exists\]\[1)\; \lim_{x\rightarrow a}\left [ f(x)+g(x) \right ]=\lim_{x\rightarrow a}\: f(x)+\lim_{x\rightarrow a}\: g(x)\]\[2)\; \lim_{x\rightarrow a}\left [ f(x)-g(x) \right ]=\lim_{x\rightarrow a}\: f(x)-\lim_{x\rightarrow a}\: g(x)\]\[3)\; \lim_{x\rightarrow a}\left [ f(x)\times g(x) \right ]=\lim_{x\rightarrow a}\: f(x)\times \lim_{x\rightarrow a}\: g(x)\]\[4)\; \lim_{x\rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}\]Limit of the constant function remain same constant. \[\lim_{x\rightarrow c}24=24\]

Undefined terms: \[\frac{0}{0},\; \; \frac{\infty }{\infty },\; \; 0^{0},\; \; \infty ^{\infty },\; \; \: \: all\: are\: called\: undefined\: form\]If we put the limit and we got undefined form then at that point limit cannot be calculated. In this case we first simplify the functions by applying some formulas and then again find the limit.

IMPORTANT FORMULAS USED IN LIMITS

\[\lim_{x\to a}\frac{x^{n}-a^{n}}{x-a}=na^{n-1}\]\[\lim_{x\to 0 }\frac{e^{x}-1}{x}=1\]\[\lim_{\lambda x\to0 }\frac{e^{\lambda x-1}}{\lambda x}=\lambda \: \: \: where\: \lambda \neq 0\]\[\lim_{x\to0 }\frac{Log(1+x)}{x}= 1,\: \lim_{x\to e }Log_{e}\: x=1\]\[\lim_{x\to0 }\frac{Log_{e}(1-x)}{x}= -1,\]

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LIMITS OF TRIGONOMETRIC FUNCTIONS

\[\lim_{x\to 0}sinx=0, \: \: \: \:\; \; \lim_{x\to 0}cosx=1\]\[\lim_{x\to 0}\: \frac{sinx}{x}=1,\: \: \: \lim_{x\to 0}\frac{x}{sinx}=1, \\  \lim_{x\to 0}\: \frac{tanx}{x}=1,\lim_{x\to0}\frac{x}{tanx}=1\]\[\lim_{x\to0}\frac{sin^{-1}x}{x}=1,\;\: \lim_{x\to0}\frac{x}{sin^{-1}x}=1\]\[\lim_{x\to0}\frac{tan^{-1}x}{x}=1,\;\: \lim_{x\to0}\frac{x}{tan^{-1}x}=1\]\[\lim_{x\to0}\frac{sinx^{o}}{x}=\frac{\pi }{180^{o}}\]\[\lim_{x\to a}\frac{sin(x-a))}{x-a}=1=\lim_{x\to a}\frac{tan(x-a))}{x-a}\]********************************************* *******\[\lim_{x\to\infty }\frac{sinx}{x}=\: 0\: =\lim_{x\to\infty }\frac{cosx}{x}\]\[\lim_{x\to\infty }\frac{sin\frac{1}{x}}{\frac{1}{x}}=\: 0\: =\lim_{x\to\infty }\frac{cosx}{x}\]

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Left Hand Limit and Right Hand Limit

If at any point c given function have two different values then at that point (c) we calculate LHL and RHL. and the limit exist if

Left Hand Limit = Right Hand Limit = f(c)\[LHL \; is\; denoted\; by\; \lim_{x\rightarrow c^{-}}f(x)\]\[RHL \; is\; denoted\; by\; \lim_{x\rightarrow c^{+}}f(x)\]\[When \; limit\; exists\; then\; \lim_{x\rightarrow c^{-}}f(x)=\lim_{x\rightarrow c^{+}}f(x)=f(c)\]\[For\; a\; function\; f(x)\; the\; limits\; \lim_{x\to a^{-}}f(x)\; and\; \lim_{x\to a^{+}}f(x),\; may\; or \; may \; not\; exists\]\[If both\; limits\; \lim_{x\to a^{-}}f(x)\; and\; \lim_{x\to a^{+}}f(x)\; exists, they \; may \; or \; may\; not\; be \; equal\]\[If\;\; \lim_{x\to a^{-}}f(x)\; \neq \; \lim_{x\to a^{+}}f(x)\; ,then\; \lim_{x\to a}\; does \; not\; exist\]\[If\;\; \lim_{x\to a^{-}}f(x)\; = \; \lim_{x\to a^{+}}f(x)\; ,then\; \lim_{x\to a}\; \; exist\; and\; is\; equal\; to\; the\; common\; limit\]*********************************************** 

CONTINUITY OF A FUNCTION:

A function is said to be continuous if its limit exist   or if  \[\lim_{x \to c}f(x)=f(c)\; \; or\; \;if \]\[Left\; \; Hand\; \; Limit = Right\; \; Hand\; \; Limit = f(c)\]

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EVERYWHERE CONTINUOUS FUNCTION

There were some functions which are always continuous in their domain. The functions which are given below are always continuous in their domain

Constant Functions

Exponential Functions

Polynomial Functions

Modulus Functions 

Rational Functions

Trigonometric Functions

Logarithmic Functions

 

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Some other formulas used in limits 

EXPONENTIAL LIMITS\[Here\; we\; use\; the\; series:\:\: \: e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+.......\infty\]\[\lim_{x\to \infty }\frac{1}{x}=0\; \; \; \; \; \; \lim_{x\to \infty }\frac{1}{x^{k}}=0\]\[\lim_{x\to 0 }(1+x)^{\frac{1}{x}}=e\]\[\lim_{x\to0 }(1+\lambda x)^{\frac{1}{x}}=e^{\lambda }\]\[\lim_{x\to\infty }(1+\frac{1}{x})^{x}=e,\: \: \lim_{x\to\infty }(1+\frac{\lambda }{x})^{x}=e^{\lambda }\]

LOGARITHMIC LIMITS \[Here\; we\; use\; the \; series:\: \: Log(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-......\infty\]\[\lim_{x\to0 }\frac{Log_{a}(1+x)}{x}= Log_{a}\: e\]\[\lim_{x\to 0 }\frac{a^{x}-1}{x}=log\: a\: \: where\: \: a> 0\] 

Differentiations

DERIVATIVE BY FIRST PRINCIPAL

\[f^{'}(x)=\lim_{ h \to0}\frac{f(x+h)-f(x)}{h}\; (OR)\; f^{'}(x)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\]

DIFFERENTIATION OF SOME IMPORTANT  FUNCTIONS

\[\frac{d}{dx}(x)=1\]

\[\frac{d}{dy}(x^{2})=2x,\; \; \; \frac{d}{dx}(x^{3})=3x^{2},\; \; \; \frac{d}{dx}(x^{4})=4x^{3}\]

\[\frac{d}{dx}x^{n}=nx^{n-1}\]

\[\frac{d}{dx}(ax+b)=a\]

\[\frac{d}{dx}\left (ax+b \right )^{3}=3(ax+b)^{3-1}\frac{d}{dx}(ax+b)=3a(ax+b)^{2}\]

\[\frac{d(constant)}{dx}=\frac{d(C)}{dx}=0\]

\[\frac{d}{dx}e^{x}=e^{x}\]

\[\frac{d}{dx}logx=\frac{1}{x}\]

\[\frac{d}{dx}\left ( \frac{1}{x} \right )=\frac{-1}{x^{2}}\]

\[\frac{d}{dx}\left ( \frac{1}{x^{2}} \right )=\frac{-2}{x^{3}}\]

\[\frac{d}{dx}\left ( \frac{1}{x^{3}} \right )=\frac{-3}{x^{4}}\]

\[\frac{d}{dx}\left ( \frac{1}{x^{n}} \right )=\frac{-n}{x^{n+1}}\]

PRODUCT RULE OF DIFFERENTIATION

\[\left ( uv \right )^{'}=u^{'}v+uv^{'}\] or \[\frac{d(uv)}{dx}=\left (\frac{du}{dx} \right )v+u\left ( \frac{dv}{dx} \right )\]

QUOTIENT RULE OF DIFFERENTIATION

\[\left ( \frac{u}{v} \right )^{'}=\frac{u'v-uv'}{v^{2}}-\] or \[\frac{d}{dx}\left ( \frac{u}{v} \right )=\frac{\left (\frac{d}{dx}u \right )v-u\left ( \frac{d}{dx}v \right )}{v^{2}}\]

DIFFERENTIATION OF SOME TRIGONOMETRIC   FUNCTIONS

\[\frac{d}{dx}sinx=cosx\]

\[\frac{d}{dx}cosx=-sinx\]

\[\frac{d}{dx}tanx=sec^{2}x\]

\[\frac{d}{dx}cotx=-cosec^{2}x\]

\[\frac{d}{dx}secx=secx\; tanx\]

\[\frac{d}{dx}cosecx=-cosecx\; cotx\]

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