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summary of this topic

Limit of Algebraic Expressions

Limit of Algebraic Expressions

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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`

Limit of functions evaluating to `oo`

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

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


Keep tapping on the content to continue learning.
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`'

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Progress

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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"

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