Limit and continuity | Learn Maths Online

Limit and continuity 

Limit of a function f(x) is said to exist as, x \to a when

 {\lim }\limits_{x \to {a^ + }} f(x) = {\lim }\limits_{x \to {a^ - }} f(x) =   finite quantity.

 

 <img src="limit.png" alt="limit">

Fundamental Theorems On Limit and continuity :

Let    {\lim }\limits_{x \to {a^{}}} f(x) = l &   {\lim }\limits_{x \to {a^{}}} f(x) = l   If l & m exists then :

(i) f (x) ± g (x) = l ± m

 

(ii) f(x). g(x) = l. m

 

(iii)  {\lim }\limits_{x \to \infty } \frac{{f(x)}}{{g(x)}} = m  provided  m \ne 0

 

(iv)  {\lim }\limits_{x \to {a^{}}} kf(x) = k {\lim }\limits_{x \to {a^{}}} f(x)   where k is a constant.

 

(v)    {\lim }\limits_{x \to {a^{}}} f[g(x)] = f[ {\lim }\limits_{x \to {a^{}}} g(x)] = f(m)provided f is continuous at        g (x) = m

Standard Limit and continuity  :

(a)  {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} = 1 and {\lim }\limits_{x \to 0} \frac{{\tan x}}{x} = {\lim }\limits_{x \to 0} \frac{{{{\tan }^{ - 1}}x}}{x} = 1 {\lim }\limits_{x \to 0} \frac{{{{\sin }^{ - 1}}x}}{x} = 1 Where x is measured in radians

 

(b)  {\lim }\limits_{x \to 0} {(1 + x)^{\frac{1}{x}}}and {\lim }\limits_{x \to 0} {(1 + \frac{1}{x})^x} both are equal to e

 

(c) {\lim }\limits_{x \to a} f(x) = 1and {\lim }\limits_{x \to a} \theta (x) = \infty  then this will show that  {\lim }\limits_{x \to a} f{(x)^{ {\lim }\limits_{x \to a} \theta (x)}} = {e^{ {\lim }\limits_{x \to a} \theta (x)[f(x) - 1]}}

 

(d)  {\lim }\limits_{x \to a} f(x) = A > 0 and   {\lim }\limits_{x \to a} \theta (x) = B (a finite quantity) then    {\lim }\limits_{x \to a} f{(x)^{ {\lim }\limits_{x \to a} \theta (x)}} = {e^z}

 

where z= ^{ {\lim }\limits_{x \to a} \theta (x)\ln f(x)} = {e^{B\ln A}} = {A^B}

 

(e)  {\lim }\limits_{x \to 0} \frac{{{a^x} - 1}}{x} = \ln a where a>0. In particular  {\lim }\limits_{x \to 0} \frac{{{e^x} - 1}}{x} = 1

Indeterminant Forms Limit and continuity 

\frac{0}{0},\frac{\infty }{\infty },0 \times \infty ,{0^\infty },{\infty ^0} etc are considered to be indeterminant values

We cannot plot  \infty   on the paper. Infinity \infty  is a symbol & not a number. It does not obey the laws of elementary algebra.

\infty +\infty =\infty

\infty ×\infty \infty

(a/\infty ) = 0 if a is finite and denominator is not defined

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a b =0,if & only if a = 0 or b = 0  and  a & b are finite.

Expansion of function like Binomial expansion, exponential & logarithmic expansion, expansion of sinx , cosx , tanx should be remembered by heart & are given below:

(i)  ex =1+x/1!+x3/3!+x4/4!……\infty

 

(ii)  ax=1+(xloga)/1!+ (xloga)2/2!+ (xloga)3/3!+ (xloga)4/4!+……….where a > 0

 

(iii)   ln(1-x)=x-x2/2+x3/3-x4/4……….    where -1 < x  1

 

(iv)  ln(1-x)=-x-x2/2-x3/3-x4/4……….     where  -1 x < 1

 

(v )  \sin x = x - \frac{{{x^3}}}{{3!}} + \frac{{{x^5}}}{{5!}} - \frac{{{x^7}}}{{7!}}.......

 

(vi) \cos x = 1 - \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}} - \frac{{{x^6}}}{{6!}}.......

 

(v)  \tan x = x + \frac{{{x^3}}}{3} + \frac{{2{x^5}}}{{5!}} - ..........

 

In the next post on Limit and continuity, IB Maths Tutors will discuss various types of limit problems, their solutions and L’ Hospital’s rule. In the meantime, you can solve these basic questions from this PDF. This PDF is for beginners only. Our Online Maths Tutors will post difficult and higher level questions in the next post on this topic

 Limits and continuity PDF.pdf

In my second post on Limit and continuity, you can learn how to solve different types of questions on limits
Here is the link

http://ibelitetutor.com/blog/how-to-solve-limit-problems/

<img src="demo.png" alt="demo">

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