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Thought-Process to Discover Knowledge

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User Guide

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jogger,

exercise.

User Guide

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User Guide

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User Guide

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User Guide

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summary of this topic

### Algebra of Limits

Voice

Voice

Home

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.

Keep tapping on the content to continue learning.
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

Progress

Progress

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

we are not perfect yet...