Show that lim[(nx)/(1+n^2(x^2))=0 for all x in R Show that lim[(nx)/(1+n^2(x^2))=0 for all x in R Then show that if a>0 then the convergence of the sequence above is uniform on the interval...

Show that lim[(nx)/(1+n^2(x^2))=0 for all x in R 

Show that lim[(nx)/(1+n^2(x^2))=0 for all x in R

Then show that if a>0 then the convergence of the sequence above is uniform on the interval [a,∞], but is not uniform on the interval [0,∞)

Asked on by assinp

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cosinusix's profile pic

cosinusix | College Teacher | (Level 3) Assistant Educator

Posted on

I disagree with the previous answer. The question is not a limit for  `nrarroo` and not a limit for  `x->oo`

 

For any value of x. x can be seen as a constant parameter

 

 

`f_x(n)=nx/(1+n^2x^2)=(1/n)x/(1/n^2+x^2)`

 

If x=0, f_x(n)=0/1=0

For any value of n f_0(n)=0 therefore lim_(n->oo)f_0(n)=0.

If `xne 0.`

`lim_(nrarroo)f_x(n)=lim_(xrarroo)(1/n)x/(1/n^2+x^2)=lim_(n->oo)(1/n)*1/x=0` (`xne 0 ` is a constant)

 

 

 

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to evaluate the limit of function `lim_(x-gtoo)(nx)/(1+n^2(x^2)),`  hence you need to factor out `x^2`  to denominator such that:

`lim_(x-gtoo) (nx)/(1+n^2(x^2)) = lim_(x-gtoo)(nx)/(x^2(1/x^2 + n^2))`

You need to reduce fraction by x such that:

`lim_(x-gtoo)(n)/(x(1/x^2 + n^2))`

Substituting oo for x yields:

`lim_(x-gtoo)(n)/(x(1/x^2 + n^2)) = n/(oo(1/oo + n^2))`

`lim_(x-gtoo)(n)/(x(1/x^2 + n^2)) = n/oo = 0`

Hence, evaluating the limit of function under given conditions yields `lim_(x-gtoo)(nx)/(1+n^2(x^2)) = 0`  .

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