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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

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Limit of a defined Function

» A function `f(x)` at `x=a` is

→ **defined by limit**: if `f(a)=0/0` and LHL = RHL

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

If the left hand and right hand limits are equal then the *limit is defined* at that input value.

If the function evaluates to indeterminate value and the limit is defined, then the function is defined by the limits.

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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

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

Given function `f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1)`. what is `f(x)|_(x=1)`?

- Substitute `x=1`
- `(1-1)/(1-1)`
- `0/0`
- all the above

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

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

- `2`
- `0/0`
- `-delta`
- `-2`

The answer is '`2`'.

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

`quad quad = color(deepskyblue)((1-delta)^2-1)/color(coral)((1-delta)-1)`

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

`quad quad = color(deepskyblue)(-delta(2-delta)) /color(coral)(-delta)`

`quad quad = 2-delta`

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

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

- `2`
- `0/0`
- `delta`
- `-2`

The answer is '`2`'.

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

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

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

`quad quad = color(deepskyblue)(delta(2+delta)) /color(coral)(delta)`

`quad quad = 2+delta`

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

Given function `f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1)`.

• `f(1)=0/0` indeterminate value

• `lim_(x->1-)f(x) = 2`

• `lim_(x->1+)f(x) = 2`

The function is defined at `x=1`.

When left-hand-limit and right-hand-limit are equal, then limit is defined.

Limit is referred as "limit of the function"

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Limit is defined: **If `lim_(x->a-)f(x)` `= lim_(x->a+)f(x)` then the limits are commonly given as `lim_(x->a) f(x)` **Function is defined by Limit: ** If `f(a) !in RR` and `lim_(x->a-)f(x)` `= lim_(x->a+)f(x)`, then the function is defined by limit at `x=a`.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

Find if the function `f(x)=color(deepskyblue)(3x^2-2x-8)/color(coral)(x-2)` is continuous or defined at `x=2`?

- Both continuous and defined
- Not defined
- Not Continuous and Defined

The answer is 'Not Continuous and Defined'

*your progress details*

Progress

*About you*

Progress

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

substitute;x

Substitute x=1

minus

1 minus 1 by 1 minus 1

by;0

0 by 0

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

2

2

0;by

0 by 0

delta

minus delta

minus 2;-2

minus 2

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

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

2

2

0;by

0 by 0

delta

delta

minus 2;-2

minus 2

The answer is "2" ;; limit x tending to 1 +, f of x ;; equals 1 + delta squared minus 1, by, 1 + delta minus 1 ;; equals 1 + 2 delta + delta squared minus 1, by, 1 + delta minus 1;; equals delta into 2 + delta, by delta;; equals 2 + delta;; equals 2

Given function f of x = x squared minus 1 by x minus 1. ;; f of 1 = 0 by 0, indeterminate value;; limit x tending to 1 minus, f of x = 2;; limit x tending to 1 +, f of x = 2;; The function is defined at x=1.

When left-hand-limit and right-hand-limit are equal, then limit is defined. Limit is referred as "limit of the function"

If the left hand and right hand limits are equal then the limit is defined at that input value. ;; If the function evaluates to indeterminate value and the limit is defined, then the function is defined by the limits.

Limit is defined : If limit x tending to a. minus, f of x equals limit x tending to a. +, f of x then limits are commonly given as limit x tending to a. f of x ;; Function is defined by limit: if f of a not in real numbers and limit x tending to a. minus, f of x equals limit x tending to a. +, f of x, then the function is defined by limit at x = a.

Find if the function f of x = 3 x squared minus 2 x minus 8 by x minus 2 is continuous or defined at x = 2?

both

Both continuous and defined

not defined

Not defined

not continuous

Not Continuous and Defined

The answer is 'Not Continuous and Defined'