Continuity of functions-IB Maths topics

Continuity of functions-

The word continuous means without any break or gap. Continuity of functions exists when our function is without any break or gap or jump . If there is any gap in the graph, the function is said to be discontinuous.

Graph of functions like sinx,cosx, secx, 1/x etc are continuous (without any gap) while greatest integer function has a break at every point(discontinuous).

1. A function f(x) is said to be continuous at x = c,  if  {\lim }\limits_{x \to c} f(x) = f(c) .

 

symbolically f is continuous at x = c if  {\lim }\limits_{x \to c - h} f(c + h) = {\lim }\limits_{x \to c - h} f(c - h) = f(c).

 

It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighborhood of x = a, not necessarily at x = a.

<img src="continuous functions.png" alt="continuous functions">

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How To Solve Limit Problems

How To Solve Limit Problems

 

In my previous post on limits, We have discussed some basic as well as advanced concepts of limits. Here we shall discuss different methods to solve limit questions. Based on the type of function, we can divide all our work into sections-:

Algebraic Limits- Problems of limits that involve algebraic functions are called algebraic limits. They can be further divided into following sections:-

Direct Substitution Method –Suppose we have to find. L = {\lim }\limits_{x \to a} f(x) we can directly substitute the value of the limit of the variable (i.e replace x=a) in the expression.

► If f(a) is finite then L=f(a)

► If f(a) is undefined then L doesn’t exist

► If f(a) is indeterminate  then this method fails

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

Example-1:- Find value of   {\lim }\limits_{x \to 2} (x²-5x+6) Read more

Limit, Continuity & Differentiability-IB Maths Topics

Limit of a function

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 Limits :

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 Limits :

(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:

\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 v is not defined

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 next post, I 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. I will post difficult and higher level questions in the next post on this topic

 LimitsExercises.pdf

In my second post on limits, 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/