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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

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

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

Welcome to **nub****trek**.

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

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

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.

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

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

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

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

Voice

Voice

Home

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`

*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

You are learning the free content, however do shake hands with a coffee to show appreciation.

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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.

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. 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.