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) anf g(x) are two functions such that $\lim_{x\rightarrow 0}\;f(x)$ and $\lim_{x\rightarrow 0}\;g(x)$ exist.

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) $\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\displaystyle\lim_{x\to 0}f(x)}{\displaystyle\lim_{x\to 0}g(x)}$

Limit of constant function remain constant

$\lim_{x\rightarrow c}24=24$

Undefined terms:

$\frac{0}{0},\;\;\frac{\infty}{\infty},\;\;0^{0},\;\;\infty^{\infty}$ all these terms are called undefined terms

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

$\lim_{x\to 0}sinx=0$

$\lim_{x\to 0}cosx=1$

$\lim_{x\to 0}\:\frac{sinx}{x}=1$

$\lim_{x\to 0}\frac{x}{sinx}=1$

$\lim_{x\to 0}\:\frac{tanx}{x}=1$
$\lim_{x\to0}\frac{x}{tanx}=1$
$\lim_{x\to0}\frac{sin^{-1}x}{x}=1$

$\lim_{x\to0}\frac{x}{sin^{-1}x}=1$

$\lim_{x\to0}\frac{tan^{-1}x}{x}=1$
$\lim_{x\to0}\frac{x}{tan^{-1}x}=1$
$\lim_{x\to0}\frac{sinx^{o}}{x}=\frac{\pi}{180^{o}}$

$\lim_{x\to a}\frac{sin(x-a))}{x-a}=1$

$\lim_{x\to a}\frac{tan(x-a))}{x-a}=1$

$\lim_{x\to\infty}\frac{sinx}{x}=\:0$

$\lim_{x\to\infty}\frac{cosx}{x}=0$

$\lim_{x\to\infty}\frac{sin\frac{1}{x}}{\frac{1}{x}}=0$

$\lim_{x\to\infty}\frac{cosx}{x}=0$

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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 exist 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 exist

If both limits $\lim_{x\to a^{-}}f(x)$ and $\lim_{x\to a^{+}}f(x)$ exist, they may not be equal.

If $\lim_{x\to a^{-}}f(x)\;\neq\;\lim_{x\to a^{+}}f(x)$ then $\lim_{x\to a}f(x)$ does not exist

If $\lim_{x\to a^{-}}f(x)\;=\;\lim_{x\to a^{+}}f(x)$ , then $\lim_{x\to a}f(x)$ exist and is equal to the common limit.

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
$e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+.......\infty$

$\lim_{x\to\infty}\frac{1}{x}=0$

$\lim_{x\to\infty}\frac{1}{x^{k}}=0$

$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=e$

$\lim_{x\to0}(1+\lambda x)^{\frac{1}{x}}=e^{\lambda}$

$\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}=e$

$\lim_{x\to\infty}\left(1+\frac{\lambda}{x}\right)^{x}=e^{\lambda}$

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

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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