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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

User Guide   

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
  think,
  figure-out, &
  learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn. continue

User Guide   

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

  nub,

  trek,

  jogger,

  exercise.

continue

User Guide    

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about. continue

User Guide    

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step. continue

User Guide    

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen. continue

User Guide    

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge. continue

summary of this topic

Basics: Limit of a function

Basics: Limit of a function

Voice  

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Value of a function


 »  Value of `f(x)`
    →  Evaluated at input `f(x)|_(x=a)` or `f(a)`
    →  Left-hand-limit `lim_(x->a-) f(x)`
    →  Right-hand-limit `lim_(x->a+) f(x)`


 »  A function `f(x)` at `x=a` is
    →  continuous: if `f(a)` = LHL = RHL
    →  defined by value: if `f(a)` is a real number
    →  defined by limit: if `f(a)=0/0` and LHL = RHL
    →  not defined: if LHL `!=` RHL and `f(a) !in RR`

Continuity of a Function at an input value

plain and simple summary

nub

plain and simple summary

nub

dummy

A function is continuous at an input value, if the function evaluated at the input equals both left-hand limit and right-hand limit of the function at that input value.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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Even when functions have a defined value at an input value, the function may be discontinuous. Limit of the function at that input value can be used to determine continuity.


Keep tapping on the content to continue learning.
Starting on learning "continuity of function at an input value". ;; Even when functions have a defined value at an input value, the function may be discontinuous. Limit of the function at that input value can be used to determine continuity.

So far, the motivation to examine limits of a function was to evaluate the function at a input value of the argument variable where the function evaluates to 'indeterminate value'.
In this topic, another motivation to examine limits is explained.

Consider `f(x)=1/color(coral)(x-1)`. What is `f(1)`?

  • `+oo`
  • `-oo`

The answer is '`+oo`', by directly substituting `x=1`.

Consider `f(x)=1/color(coral)(x-1)`. What is `lim_(x->1+)f(x)`?

  • `+oo`
  • `-oo`

The answer is '`oo`', because
 `lim_(x->1+)f(x)`
 `quad quad = 1/color(coral)(1+delta-1) `
 `quad quad = 1/color(coral)(delta) = 1/0 = oo`

Consider `f(x)=1/color(coral)(x-1)`. What is `lim_(x->1-)f(x)`?

  • `+oo`
  • `-oo`

The answer is '`-oo`', because
 `lim_(x->1-)f(x)`
 `quad quad = 1/color(coral)(1-delta-1) `
 `quad quad = 1/color(coral)(-delta)`
 `quad quad = -1/0 = -oo`

For the function `f(x)=1/color(coral)(x-1)`,
 •  `f(a) = oo`
 •  `lim_(x->1+)f(x) = oo`
 •  `lim_(x->1-)f(x) = -oo`

The plot of the function is given in the figure.graph of function 1/(x-1) What does this tell about the function at `x=1`?

  • function is not continuous
  • does not have any significance

The answer is 'function is not continuous'.
For a value less than `x=1`, the function is `-oo`. And at `x=1`, the function becomes `oo`

A function `f(x)` at a given input value `x=a` is continuous if all the three are equal
`f(x)|_(x=a)`
`quad quad = lim_(x->a-) f(x)`
`quad quad = lim_(x->a+) f(x)`

What does the word 'continuous' mean?

  • unbroken and continue from one side to another without pause in between
  • broken into pieces

The answer is 'unbroken and continue from one side to another without pause in between'.

What is the term used to describe a function at an input value, if value of the function, left-hand-limit, and right-hand-limit are equal ?

  • Practice Saying the Answer

The answer is 'continuous'

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Continuity of a Function: A function `f(x)` is continuous at `x=a` if all the following three have a defined value and are equal

 •  Evaluated at the input value `f(x)|_(x=a)`

 •  left-hand-limit `lim_(x->a-) f(x)`

 •  right-hand-limit `lim_(x-a+) f(x)`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Given function `f(x) = 2x^2`, is it continuous at x=0?

  • Yes, Continuous.
  • No, Not continous

The answer is 'Yes, Continuous'. Evaluate the three values of the function and they are equal.

your progress details

Progress

About you

Progress

So far, the motivation to examine limits of a function was, to evaluate the function at a input value of the argument variable where the function evaluates to 'indeterminate value' ;; In this topic, another motivation to examine limits is explained.
Consider f of x equal 1 by x minus 1. What is f of 1?
plus
plus infinity
minus
minus infinity
The answer is "plus infinity", by directly substituting x = 1
Consider f of x equal 1 by x minus 1. What is limit x tending to 1 plus f of x?
plus
plus infinity
minus
minus infinity
The answer is "plus infinity", because, ;; limit x tending to 1 plus f of x ;; equals 1 by 1 plus delta minus 1;; equals 1 by delta ;; equals 1 by 0 ;; equals infinity.
Consider f of x equal 1 by x minus 1. What is limit x tending to 1 minus f of x?
plus
plus infinity
minus
minus infinity
The answer is "minus infinity" limit x tending to 1 minus f of x ;; equals 1 by 1 minus delta minus 1;; equals 1 by minus delta ;; equals minus 1 by 0 ;; equals minus infinity.
Consider f of x equal 1 by x minus 1. ;; f of a = infinity ;; limit x tending to 1 plus f of x equals infinity ;; limit x tending to 1 minus f of x equals minus infinity;; The plot of the function is given in the figure. ;; What does this tell about the function at x=1?
function;continuous
function is not continuous
does;have;any;significance
does not have any significance
The answer is "function is not continuous". For a value less than x = 1, the function is minus infinity. And at x=1, the function becomes infinity.
A function f of x at a given input value x=a, is continuous if all the three are equal ;; f of x at x=a ;; limit x tending to a. minus f of x ;; limit x tending to a. plus f of x
What does the word 'continuous' mean?
unbroken;continue;from
unbroken and continue from one side to another without pause in between
broken
broken into pieces
The answer is 'unbroken and continue from one side to another without pause in between'.
What is the term used to describe a function at an input value, if value of the function, left-hand-limit, and right-hand-limit are equal ?
continuous
The answer is 'continuous'
A function is continuous at an input value, if the function evaluated at the input, equals, both left-hand limit, and right-hand limit, of the function at that input value.
Continuity of a Function: A function f of x is continuous at x=a if all the following three have a defined value and are equal;; Evaluated at the input value ;; left-hand-limit ;; right-hand-limit ;;
Given function f of x equals 2 x squared, is it continuous at x=0?
yes;s
Yes, Continuous.
no;not
No, Not continous
The answer is 'Yes, Continuous'. Evaluate the three values of the function and they are equal.

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