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Algebra of Limits

Algebra of Limits

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Change of Variable in a Limit


 »  variable can be substituted
when value is not any of the forms of `0/0`
    →  `lim_(x->a)f(x)` ` = lim_(y->g(a))f(g^(-1)(y))`

Limit with change of variable

plain and simple summary

nub

plain and simple summary

nub

dummy

The variable in a limit can be changed.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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Algebra of limit with change of variable is explained.


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Starting on learning "Limit with change of variables". Algebra of limit with change of variable is explained.

Given
`lim_(x->0) (sin x)/x = 1` ;
What is `lim_(x->0) (sin(x^2))/x`?

  • cannot find the limit
  • take `y=x^2` and rearrange the function

The answer is 'take `y=x^2` and rearrange the function'

`lim_(x->0) (sin(x^2))/x`
`quad quad = lim_(x->0) x (sin(x^2))/(x^2)`
`quad quad = lim_(x->0) x xx lim_(y->0) (sin y)/y`
where `y=x^2` and `lim_(x->0)` changes to `lim_(y->0)` by the definition of y. `quad quad = 0 xx 1`
`quad quad = 0`

Note: If, in another case, `y=cos x` then `lim_(x->0)` changes to `lim_(y->1)`, as `y=cos 0 = 1`.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Change of variable in a Limit: Given that `y=g(x)` exists at `x=a`. Then
`lim_(x->a)f(x)`
`quad quad = lim_(y->g(a))f(g^(-1)(y))`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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given that limit x tending to 0, sine x by x = 1;; what is limit x tending to 0, sine x squared by x?
cannot;find
cannot find the limit
take;y;why;x;squared
take y = x squared and rearrange the function
The answer is "take y = x squared and rearrange the function" ;; limit x tending to 0, sine x squared by x ;; equals limit x tending to 0 x into sine x squared by x squared ;; equals limit x tending to 0 x into limit y tending to 0 sine y by y ;; where y equals x squared, and limit x tending to 0 changes to limit y tending to 0 by the definition of y. ;; equals 0 multiplied 1 ;; equals 0 ;; Note: If, in another case, y = cos x then limit x tending to 0 changes to limit y tending to 1 as y, = cos 0, = 1.
The variable in a limit can be changed.
Change of variable in a Limit: Given that y = g of x exists at x=a. Then limit x tending to a f of x, = limit y tending to, g of a, f of, g inverse, of y

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