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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.

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.

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.

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.

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.

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.

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.

summary of this topic

### Basics: Limit of a function

Voice

Voice

Home

A function Defined by Limits

»  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

### Function defined at an input value

plain and simple summary

nub

plain and simple summary

nub

dummy

A function is defined at an input value, if the function evaluates to a definite value OR if the left-hand-limit and right-hand-limits are equal.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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The two conditions, to figure out if a function is defined at an input value or not, are explained.

Keep tapping on the content to continue learning.
Starting on learning "function defined at an input value". ;; The two conditions, to figure out if a function is defined at an input value or not, are explained.

Under which of the following conditions, a function f(x) is defined at the input value x=a?

• when f(a) evaluates to a defined value
• when f(a) is oo
• when f(a) is 0/0
• all the above

The answer is 'when f(a) evaluates to a defined value'.

If f(a) evaluates to oo or 0/0, then the function need to be further examined.

If the left-hand-limit and right-hand-limit take a defined value and are equal, then the function is defined at that input value.

This is understood from

•  In an application scenario, the value is not exact and so the function will evaluate to a stable value given by limits.

•  In an abstract scenario, the expected value at that input value can be taken to be the value of the function.

What does the word 'defined' mean?

• give exactly; state or describe without any ambiguity
• extremely loud; blaring in high pitch

The answer is 'give exactly; state or describe without any ambiguity'.

Difference between Defined and Continuous :
A function is defined if it evaluates to a definite value OR when is the function evaluates to indeterminate value 0/0, then left-hand-limit and right-hand-limit are equal.

A function is continuous only when it evaluates to a definite value at the input value AND also left-hand-limit and right-hand-limit are equal to the definite value at the input value.

•  For a function to be continuous at a input value, it has to be defined at that input value.

•  If a function is defined at an input value, it does not imply it is continuous. It may be or may not be continuous.

•  If a function is not defined at an input value, it implies it is not continuous.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Function is defined: Given a function f(x), the function is defined at x=a
•  if f(a) is a defined value. OR

•  if f(a) is an indeterminate value 0/0 or undefined large oo, then lim_(x->a+)f(x)=lim_(x->a-)f(x)

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

If a function is not continuous at an input value, can the function be defined at that input value?

• Yes.
• No.

The answer is 'Yes.'. Theoretically a function can have a value f(a) different to the limits.

Progress

Progress

Under which of the following conditions, a function f of x is defined at the input value x=a?
evaluates;defined;value
when f of a evaluates to a defined value
infinity
when f of a is infinity
0;by
when f of a is 0 by 0
all;above
all the above
The answer is "when f of a evaluates to a defined value"
If f of a evaluates to infinity or 0 by 0, then the function need to be further examined. ;; If the left-hand-limit and right-hand-limit take a defined value and are equal, then the function is defined at that input value.;; This is understood from : In an application scenario, the value is not exact and so the function will evaluate to a stable value given by limits.;; In an abstract scenario, the expected value at that input value can be taken to be the value of the function.
A function is defined at an input value, if the function evaluates to a definite value, OR, if the left-hand-limit and right-hand-limits are equal.
What does the word 'defined' mean?
give;exactly;state;describe;any;ambiguity
give exactly; state or describe without any ambiguity
extremely;loud;blaring;high
extremely loud; blaring in high pitch
The answer is 'give exactly; state or describe without any ambiguity'.
Function is defined: Given a function f of x , the function is defined at x=a ;; if f of a is a defined value.; OR ; if f of a is an indeterminate value 0 by 0 or undefined large infinity, then limit x tending to a. + f of x , = , limit x tending to a. minus f of x
Difference between Defined and Continuous : A function is defined if it evaluates to a definite value OR when is the function evaluates to indeterminate value 0 by 0, then left-hand-limit and right-hand-limit are equal. ;; A function is continuous only when it evaluates to a definite value at the input value AND also left-hand-limit and right-hand-limit are equal to the definite value at the input value.
For a function to be continuous at a input value, it has to be defined at that input value. ;; If a function is defined at an input value, it does not imply it is continuous. It may or may not be continuous. ;; If a function is not defined at an input value, it implies it is not continuous.
If a function is not continuous at an input value, can the function be defined at that input value?
yes;s
Yes.
no
No.
The answer is "Yes". Theoretically a function can have a value f of a different to the limits.

we are not perfect yet...