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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 Expressions evaluating to `oo`

» Organize the sub-expressions to the following

→ `a/oo = 0`

→ `oo+-a = oo`

→ `oo xx a = oo` when `a != 0`

→ `oo xx oo = oo`

→ `oo^n = oo` when `n != 0`

→ `lim_(x->oo) x/x =1`

→ `lim_(x->-oo) x/x =1`

→ `lim_(x->oo) a^x` `=0 text( if ) a<1`

`= oo text( if ) a>1`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

When evaluating limits to infinity or minus infinity, simplify to known results.

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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Finding limit of standard ratios evaluating to `oo/oo` or `oo-oo` is explained with examples.

Starting on learning "Limits of functions evaluating to infinity". ;; Finding limit of standard ratios evaluating to infinity by infinity, or, infinity minus infinity is explained with examples.

What is the values of function `f(x)=3x^2+5x-2` at `x=oo`?

- `oo`
- `-2`
- `0`

The answer is '`oo`'

By substitution `x=oo`

`(3x^2+5x-2)`

`quad quad = (3(oo)^2+5 oo - 2)`

`quad quad = oo`

as `oo^2=oo`; `n oo = oo`; and `oo +- a = oo`

What is the values of function `f(x)=1/(3x^2+5x-2)` at `x=oo`?

- `oo`
- `-2`
- `0`

The answer is '`0`'

By substitution `x=oo`

`1/(3x^2+5x-2)`

`quad quad = 1/(3(oo)^2+5 oo - 2)`

`quad quad = 1/oo`

`quad quad = 0`

as `1/infinity = 0`.

What is the value of function `f(x)=3x^2-5x-2` at `x=oo`?

- `oo-oo`
- `1`
- `0`

The answer is '`oo - oo`'

By substitution `x=oo`

`3x^2-5x-2`

`quad quad = 3 oo^2 - 5 oo -2 `

`quad quad = oo - oo`

as `oo^2=oo`; `n oo = oo`; and `oo +- a = oo` Note: The limit to this function is explained after few pages.

What is the value of `oo - oo`?

- `oo`
- `1`
- `0`
- indeterminate value

The answer is 'indeterminate value'

The equivalence can be explained with

`oo - oo`

`quad quad = (1/0) - (1/0)`

`quad quad = (1-1)/0`

`quad quad = 0/0`

What is the value of function `f(x)=color(deepskyblue)(3x^2+5x-2)/color(coral)(x^2+x-2)` at `x=oo`?

- `oo/oo`
- `1`

The answer is '`oo/oo`'

By substitution `x=oo`

`color(deepskyblue)(3x^2+5x-2)/color(coral)(x^2+x-2)`

`quad quad = color(deepskyblue)(3(oo)^2+5 oo - 2)/color(coral)(oo^2+oo-2)`

`quad quad = color(deepskyblue)(oo)/color(coral)(oo)`

as `oo^2=oo`; `n oo = oo`; and `oo +- a = oo`

Note: The limit to this function is explained after few pages.

What is the value of `oo/oo`?

- `oo`
- `1`
- `0`
- indeterminate value

The answer is 'indeterminate value'

The equivalence can be explained with

`oo/oo`

`quad quad = (1/0) -:(1/0)`

`quad quad = 1/0 xx 0/1`

`quad quad = 0/0`

What forms of expressions evaluate to indeterminate values when computing limit for `oo` or `-oo`?

- `oo xx oo` and `oo + oo`
- `oo -: oo` and `oo - oo`

The answer is '`oo -: oo` and `oo - oo`'

When we encounter `oo -: oo` or `oo - oo`, convert the expression to one of the following forms given on left hand side

`lim_(x->oo) x/x = 1`

`lim_(x->-oo) x/x = 1`

`a/oo = 0`

`oo^n=oo`

`n oo = oo`

`oo +- a = oo`

Limit of function `f(x)=(3x^2-5x-2)` at `x=oo`

The function evaluates to `oo-oo` at `x=oo`

The limit of the function is

`lim_(x->oo) (3x^2-5x-2)`

`quad quad = lim_(x->oo) x^2(2-5/x - 2/x^2) `

`quad quad = lim_(x->oo) x^2 `

`quad quad quad quad xx lim_(x->oo) (2-5/x - 2/x^2)`

`quad quad = oo^2 xx (2-0-0)`

`quad quad = oo`

Function `f(x)=color(deepskyblue)(3x^2+5x-2)/color(coral)(x^2+x-2)` at `x=oo`

The function evaluates to `oo/oo` at `x=oo`

The limit of the function is

`lim_(x->oo) color(deepskyblue)(3x^2+5x-2)/color(coral)(x^2+x-2)`

`quad quad = lim_(x->oo) color(deepskyblue)(x^2(3+5/x-2/x^2))/color(coral)(x^2(1+1/x-2/x^2)) `

`quad quad = lim_(x->oo) color(deepskyblue)(x^2)/color(coral)(x^2) `

`quad quad quad quad xx lim_(x->oo)color(deepskyblue)(3+5/x-2/x^2)/color(coral)(1+1/x-2/x^2) `

`quad quad = [lim_(x->oo) color(deepskyblue)(x)/color(coral)(x)]^2 xx color(deepskyblue)(3+0-0)/color(coral)(1+0-0)`

`quad quad = 1^2 xx 3`

`quad quad = 3`

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Evaluating limits to `oo` or `-oo`: **Simplify the numerical expressions to one of the following

`lim_(x->oo) x/x = 1`

`lim_(x->-oo) x/x = 1`

`a/oo = 0`

`oo +- a = oo`

`n oo = oo` where `n!=0`

`oo xx oo = oo` or

`oo^n=oo` where `n!=0`

And avoid indeterminate values `oo/oo`, `oo-oo`, `0 xx oo`, and `oo^0` .

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

Find the limit of the function `lim_(x->oo) (x+3)/(5x+4)`

- `1/5`
- `5`
- `oo`
- `0`

The answer is '`1/5`'

*your progress details*

Progress

*About you*

Progress

What is the values of function f of x equals 3 x squared + 5 x minus 2 at x = infinity?

infinity

infinity

minus;2

minus 2

0

0

The answer is "infinity". ;;By substitution x= infinity;; 3 x squared + 5 x minus 2 ;; equals ;; 3 infinity squared + 5 times infinity minus 2;; equals infinity ;; as infinity squared = infinity; n times infinity = infinity;; infinity plus of minus a constant = infinity.

What is the values of function f of x equals, 1 divided by, 3 x squared + 5 x minus 2 at x = infinity?

infinity

infinity

minus

minus 2

0

0

The answer is "0". ;; By substitution x = infinity;; 1 divided by 3 x squared + 5 x minus 2 ;; equals ;; 1 divided by 3 infinity squared + 5 times infinity minus 2;; equals 1 by infinity ;; equals 0 ;; as 1 by infinity = 0.

What is the value of function f of x = 3 x squared minus 5 x minus 2 at, x = infinity?,

infinity;minus

infinity minus infinity

1

1

0

0

The answer is "infinity minus infinity". ;; By substitution x = infinity;; 3 x squared minus 5 x minus 2 ;; equals 3 infinity squared minus 5 times infinity minus 2;; equals infinity minus infinity;; Note: The limit to this function is explained after few pages.

What is the value of infinity minus infinity?

infinity

infinity

1

1

0

0

indeterminate;value

indeterminate value

The answer is "indeterminate value" ;; The equivalence can be explained with ;; infinity minus infinity ;; equals 1 by 0 minus 1 by 0 ;; equals 1 minus 1, by 0;; equals 0 by 0

What is the value of function f of x = 3 x squared + 5 x minus 2, by x squared + x minus 2 at x = infinity?

infinity

infinity by infinity

1

1

The answer is "infinity by infinity". ;; By substitution x = infinity;; 3 x squared + 5 x minus 2 , by x squared + x minus 2;; equals 3 infinity squared + 5 times infinity minus 2, divided by, infinity squared + infinity minus 2;; equals infinity by infinity;; as infinity squared = infinity, n times infinity = infinity, and, infinity plus or minus a constant = infinity ;; Note: The limit to this function is explained after few pages.

What is the value of infinity by infinity?

infinity

infinity

1

1

0

0

indeterminate;value

indeterminate value

The answer is "indeterminate value". ;; the equivalence can be explained with ;; infinity by infinity;; equals 1 by 0 divided by 1 by 0;; equals 1 by 0 multiplied by 0 by 1;; equals 0 by 0

What forms expressions evaluate to indeterminate values when computing limit for infinity or minus infinity

multiplied;plus;times

infinity multiplied infinity and infinity plus infinity

divided;minus

infinity divided by infinity and infinity minus infinity

The answer is "infinity divided by infinity and infinity minus infinity".

when we encounter, infinity divided by infinity, or, infinity minus infinity convert the expression to one of the following forms given on left hand side;; limit x tending to infinity x by x, = 1;; limit x tending to minus infinity x by x = 1;; a. by infinity = 0;; infinity power n = infinity;; infinity plus or minus a. = infinity

Limit of funcion f of x = 3 x squared minus 5 x minus 2, at x=infinity;; The function evaluates to infinity minus infinity at x = infinity;; the limit of the function is ;; limit x tending to infinity, 3 x squared minus 4 x minus 2;; equals limit x tending to infinity, x squared multiplied 2 minus 5 by x minus 2 by x squared ;; equals limit x tending to infinity, x squared multiplied limit x tending to infinity 2 minus 5 by x minus 2 by x squared;; equals infinity squared multiplied 2 minus 0 minus 0;; equals 0

function f of x =, 3 x squared + 5 x minus 2, divided by, x squared + x minus 2 at x, at x = infinity ;; The function evaluates to infinity by infinity at x= infinity ;; The limit of the function is limit x tending to infinity, 3 x squared + 5 x minus 2, divided by, x squared + x minus 2 at x;; equals limit x tending to infinity, x squared multiplied, 3 + 5 by x minus 2 by x squared, divided by x squared multiplied, 1 + 1 by x minus 2 by x squared ;; equals limit x tending to infinity, x squared by x squared ; limit x tending to infinity, 3 + 5 by x minus 2 by x squared, divided by, 1 + 1 by x minus 2 by x squared;; equals limit x tending to infinity, x by x whole squared, multiplied 3+ 0 minus 0 divided by 1 plus 0 minus 0;; equals 1 squared multiplied 3;; equals 3.

When evaluating limits to infinity or minus infinity, simplify to known results.

Evaluating limits to infinity or minus infinity: Simplify the numerical expressions to one of the following, limit x tending to infinity x by x, = 1;; limit x tending to minus infinity x by x = 1;; a. by infinity = 0;; infinity plus or minus a. = infinity ;; n times infinity = infinity ;; infinity multiplied infinity = infinity ;; infinity power n = infinity ;; And avoid indeterminate infinity by infinity, infinity minus infinity, 0 times infinity, and infinity power 0.

Find the limit of the function limit x tending to infinity x+3 divided by 5x + 4.

1;by

1 by 5

5

5

infinity

infinity

0

0

The answer is "one by 5"