Featured Posts

CBSE Assignments class 09 Mathematics

  Mathematics Assignments & Worksheets  For  Class IX Chapter-wise mathematics assignment for class 09. Important and useful extra questions strictly according to the CBSE syllabus and pattern with answer key CBSE Mathematics is a very good platform for the students and is contain the assignments for the students from 9 th  to 12 th  standard.  Here students can find very useful content which is very helpful to handle final examinations effectively.  For better understanding of the topic students should revise NCERT book with all examples and then start solving the chapter-wise assignments.  These assignments cover all the topics and are strictly according to the CBSE syllabus.  With the help of these assignments students can easily achieve the examination level and  can reach at the maximum height. Class 09 Mathematics    Assignment Case Study Based Questions Class IX 

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

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

************************************************* 

LIMITS OF TRIGONOMETRIC FUNCTIONS

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

********************************************* *******

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  \[\lim_{x \to c}f(x)=f(c)\; \; or\; \;if \]\[Left\; \; Hand\; \; Limit = Right\; \; Hand\; \; Limit = f(c)\]

*******************************************

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

 

******************************************
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

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

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

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

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

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

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

🙏


THANKS FOR YOUR VISIT
PLEASE COMMENT BELOW

 

Comments

Breaking News

Popular Post on this Blog

Lesson Plan Maths Class 10 | For Mathematics Teacher

Theorems on Quadrilaterals Ch-8 Class-IX

Lesson Plan Math Class X (Ch-13) | Surface Area and Volume

SUBSCRIBE FOR NEW POSTS

Followers