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

IB Mathematics (part-2)-Domain and Range of a function

In IB Mathematics both HL and SL, functions are one of the most important areas because they lie at the heart of much of mathematical analysis.here I am discussing domain and range of a function

Domain of a Function

Suppose I say that f is a real function.This means that for real input, the output should be real. For example

 F(x)=√x

if F is real, then x can only take non-negative values because only then the output will be real. Set of all real values of R is called domain

“Domain is the set of all possible inputs for which the output is real ”

In some cases, x is defined explicitly. for example,

                                     y=f(x)

                                       =x²;    1<x>2 here domain is defined explicitly as (1,2)

If no domain is mentioned explicitly, the domain will be assumed to be such that “F” produces real output

These are a few examples

  1. y=f(x)=√x
    Domain D= x≥0

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