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 a Continuous Function

» A function `f(x)` at `x=a` is

→ **continuous**: if `f(a)` = LHL = RHL

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

Function is *continuous* if all three values are equal.

*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 continuous function is discussed. The condition under which a function is continuous is illustrated with examples.

Starting on learning "limit of continuous functions". ;; With an example, calculation of limits for a continuous function is discussed. The condition under which a function is continuous is illustrated with examples.

Given function `f(x)=1/(x^2+1)`. what is `f(x)|_(x=1)`?

- Substitute `x=1`
- `1/(1+1)`
- `1/2`
- all the above

The answer is 'All the above'. The options provide the steps to evaluate `f(1)`.

Given function `f(x)=1/(x^2+1)`. what is left-hand-limit `lim_(x->1-)f(x)`?

- Substitute `x=1`
- Substitute `x=1+delta`
- Substitute `x=1-delta`
- all the above

The answer is 'Substitute `x=1-delta`'.

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

`quad quad = 1/((1-delta)^2+1)`

`quad quad = 1/(1-2delta+delta^2+1)`

`quad quad = 1/(2-2delta+delta^2)`

`quad quad = 1/2 quad quad quad quad` (substituting `delta=0`)

Given function `f(x)=1/(x^2+1)`. what is right-hand-limit `lim_(x->1+)f(x)`?

- Substitute `x=1`
- Substitute `x=1+delta`
- Substitute `x=1-delta`
- all the above

The answer is 'Substitute `x=1+delta`'.

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

`quad quad = 1/((1+delta)^2+1)`

`quad quad = 1/(1+2delta+delta^2+1)`

`quad quad = 1/(2+2delta+delta^2)`

`quad quad = 1/2 quad quad quad quad` (substituting `delta=0`)

Given function `f(x)=1/(x^2+1)`.

• `f(1)=1/2`

• `lim_(x->1-)f(x) = 1/2`

• `lim_(x->1+)f(x) = 1/2`

The function is continuous at `x=1`.

Note: Though, as part of the topic, the functions that are discontinuous or indeterminate or non-differentiable are mainly handled, finding limits is not only for those kind of functions. Any function at any point can be evaluated for limits.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Function is continuous: ** if `f(a)` `= lim_(x->a-)f(x)` `=lim_(x->a+)f(x)`, then the function is continuous at `x=a`.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

Given function `f(x) = sin x`, is it continuous at input values `x= 0, pi/2, pi`?

- Continuous at `0` and `pi/2` only
- Continuous at all three input values
- Continuous at only `0`
- not continuous at any one of the given values

The answer is 'Continuous at all three input values'

*your progress details*

Progress

*About you*

Progress

Given function f of x = 1 by x squared + 1. What is f of x at x =1?

substitute;x;equal

Substitute x=1

plus;1+1;+

1 by 1 + 1

2;half

1 by 2

all;above

all the above

The answer is "All the above" The options provide the steps to evaluate f of 1.

Given function f of x = 1 by x squared + 1. What is left-hand-limit limit x tending to 1 minus, f of x?

1

Substitute x=1

plus;+

Substitute x=1+delta

minus;-

Substitute x=1 minus delta

all;above

all the above

The answer is "Substitute x = 1 minus delta" ;; limit x tending to 1 minus, f of x ;; equals 1 by 1 minus delta squared +1 ;; equals 1 by 1 minus 2 delta + delta squared + 1 ;; equals 1 by 2 minus 2 delta + delta squared ;; equals 1 by 2, by substituting delta equals 0.

Given function f of x = 1 by x squared + 1. What is right hand limit limit x tending to 1 +, f of x?

1

Substitute x=1

plus;+

Substitute x=1+delta

minus;-

Substitute x=1 minus delta

all;above

all the above

The answer is "substitute x = 1 + delta" ;; limit x tending to 1 +, f of x ;; equals 1 by 1 + delta squared, +1 ;; equals 1 by 1 + 2 delta + delta squared + 1 ;; equals 1 by 2 + 2 delta + delta squared ;; equals 1 by 2, by substituting delta equals 0.

Given function f of x = 1 by x squared + 1. ;; f of 1 = 1 by 2;; limit x tending to 1 minus, f of x = 1 by 2;; limit x tending to 1 +, f of x = 1 by 2;; The function is continuous at x=1. ;; Note: Though, as part of the topic, the functions that are discontinuous or indeterminate or non-differentiable are mainly handled, finding limits is not only for those kind of functions. Any function at any point can be evaluated for limits.

Function is continuous if all three values are equal.

Function is continuous: if f of a equals, left-hand-limit, equals right-hand-limit, then the function is continuous at x=a.

Given function f of x = sine x. Is it continuous at input values x equals 0, pi by 2, pi?

only

continuous at 0 and pi by 2 only

all;three;input

Continuous at all three input values

only 0

Continuous at only 0

not;given;any;

not continuous at any one of the given values

The answer is 'Continuous at all three input values'