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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.continue
mathsLimit of a functionLimit of Algebraic Expressions

Limit of functions evaluating to `oo`

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



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

  • `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 value of function `f(x)=1/(3x^2+5x-2)` at `x=oo`?

  • `oo`
  • `-2`
  • `0`
  • `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`
  • `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`
`quad quad = 0/0`

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
  • 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`
  • `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)`
`quad quad = 0/0`

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
  • 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`
  • `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`

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

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` .

Solved Exercise Problem:

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

  • `1/5`
  • `1/5`
  • `5`
  • `oo`
  • `0`

The answer is '`1/5`'

                            
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