Calculus of a Single Variable, Chapter 8, 8.7, Section 8.7, Problem 29

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