Server Not Reachable. *This may be due to your internet connection or the nubtrek server is offline.*

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

Limit of Piecewise function

» A function `f(x)` at `x=a` is

→ **not continuous and not defined**: if LHL `!=` RHL

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

If the function evaluates to a real number, then the * function is defined* at that input value. The limit may not be defined.

*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.

*To stop this message from appearing, please choose an option and make a payment.*

With an example, calculation of limits for a piecewise function is discussed. The conditions under which a function is defined are illustrated with examples.

Starting on learning "Limit of piecewise functions". ;; With an example, calculation of limits for a piecewise function is discussed. The conditions under which a function is defined are illustrated with examples.

Given function

`f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}`

What is `f(x)|_(x=1)`?

- `0.5`
- `-1`
- `+1`

The answer is '`1`', as given by definition of the function.

Given function

`f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}`

what is left-hand-limit `lim_(x->1-)f(x)`?

- `0.5`
- `-1`
- `+1`

The answer is '`0.5`'

`lim_(x->1-)f(x)`

`quad quad = f(x)|_(x=1-delta)`

`quad quad = 0.5 quad quad quad quad` (as `1-delta<1`)

Given function

`f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}`

what is right-hand-limit `lim_(x->1+)f(x)`?

- `0.5`
- `-1`
- `+1`

The answer is '`1`'

`lim_(x->1+)f(x)`

`quad quad = f(x)|_(x=1+delta)`

`quad quad = 1 quad quad quad quad` (as `1+delta>1`)

Given function

`f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}`

• `f(1)=1`

• `lim_(x->1-)f(x) = 0.5`

• `lim_(x->1+)f(x) = 1`

The function is not continuous at `x=1`. But, the function is defined by the given definition of the function.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Function is defined by value: ** If `f(a) in RR`, and `lim_(x->a-)f(x)` `!= lim_(x->a+)f(x)`, then function is not continuous but defined by the definition of the function.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

Given function

`f(x)={(2x, quad text(for)quad x>2),(x^2, quad text(for)quad x=2),(|x|+2, quad text(for)quad x<2):}`

Examine the function at `x=2`.

- continuous
- defined by limit
- defined by value
- All the above

The answer is 'All the above'.

*your progress details*

Progress

*About you*

Progress

Given function f of x equals, 1 for x greater than equal 1, 0 point 5 for x less than 1. What is f of x at x equal 1?

0;point;5

0 point 5

minus

minus 1

plus

plus 1

The answer is ' 1 ', as given by definition of the function.

Given function f of x equals, 1 for x greater than equal 1, 0 point 5 for x less than 1. What is left-hand-limit limit x tending to 1 minus, f of x?

0;point;5

0 point 5

minus

minus 1

plus

plus 1

The answer is "0.5". ;; limit x tending to 1 minus, f of x ;; equals f of x at x = 1 minus delta ;; equals 0 point 5, as 1-delta is less than 1.

Given function f of x equals, 1 for x greater than equal 1, 0 point 5 for x less than 1. What is right-hand-limit limit x tending to 1 +, f of x?

0;point;5

0 point 5

minus

minus 1

plus

plus 1

The answer is "1". ;; limit x tending to 1 +, f of x ;; equals f of x at x = 1 + delta ;; equals 1, as 1 plus delta is greater than 1.

Given function f of x equals, 1 for x greater than equal 1, 0 point 5 for x less than 1. ;; f of 1 = 1;; limit x tending to 1 minus, f of x = 0 point 5 ;; limit x tending to 1 +, f of x = 1 ;; The function is not continuous at x = 1. But the function is defined bt the given definition of the function.

If the function evaluates to a real number, then the function is defined at that input value. The limit may not be defined.

Function is defined by value: If f of a in real numbers, and limit x tending to a. minus, f of x, not equal to, limit x tending to a. +, f of x , then function is not continuous but defined by the definition of the function.

Given function f of x equals, 2x for x greater than 2, x squared for x equals 2, mod x plus 2 for x less than 2. examine the function at x = 2

continuous

continuous

limit

defined by limit

value

defined by value

all;above

All the above

The answer is "All the Above".