Following 2 questions are from a book at a point where L’Hopital’s Rule, Squeeze Theorem etc. have not been discussed and limits (A) and (B) as given below are to be evaluated by simple methods like algebraic simplification etc.
1. Int. of (xlogx)dx from 0 to 1. Indefinite Int. comes to [x^2/2*logx-x^2/4].
Applying limits, Def. Int. =(1/2log1-1/4) – [lim as x->0 of(x^2/2*logx-0)] = -1/4, which is the required answer, if I take limit of (x^2/2*logx)=0 as x->0 ……(A).
2. Int. of (x^2*e^-x)dx from 0 to infinity. Indefinite Int. comes to [e^-x(-x^2-2x-2)].
Applying limits, Def. Int.= lim as x->infinity, of [(-x^2-2x-2)/ e^x] – [lim as x->0 of [ (-x^2-2x-2) /e^x *] = 2, which is the required answer, if I take limit of [1/e^x(-x^2-2x-2]=0 as x->infinity ……(B).
Is it possible to deduce these limits without L’Hopital’s Rule, Squeeze Theorem etc. ?
3 answers