`lim_(x->oo)x/sqrt(x+1)` Evaluate the limit, using L’Hôpital’s Rule if necessary.

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Given to solve

`lim_(x->oo) (x/(sqrt(x+1)))`

as `x-> oo` we get `(x/(sqrt(x+1))) = oo/oo ` form

so upon applying the L 'Hopital rule we get the solution as follows,

as for the general equation it is as follows

`lim_(x->a) f(x)/g(x)= 0/0` or `(+-oo)/(+-oo)` then by using the L'Hopital Rule we get  the solution with the  below form.

`lim_(x->a) (f'(x))/(g'(x))`

 

so , now evaluating

`lim_(x->oo) (x/(sqrt(x+1)))`

=` lim_(x->oo) ((x)')/((sqrt(x+1))')`

but ,

`(sqrt(x+1))' = (1/sqrt(x+1))(1/2) =(1/(2sqrt(x+1)))`

so,

`lim_(x->oo) ((x)')/((sqrt(x+1))')`

=`lim_(x->oo) (1)/((1/(2sqrt(x+1))))`

=`lim_(x->oo) (2sqrt(x+1))`

by plugging the value of `x= oo` we get

= `(2sqrt(oo+1))`

= `2sqrt(oo)`

=`oo`

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