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. ?

Yes, it is possible to evaluate these limits without using L'Hopital's Rule or the Squeeze Theorem. Here's how you can approach the two problems:

1. To find the limit in equation (A), we need to evaluate the limit as x approaches 0 of (x^2/2 * log(x)).

We can rewrite this expression as (x * x/2 * log(x)). Now, notice that as x approaches 0, both x and x/2 approach 0. Therefore, we can rewrite the limit as the product of two limits:

lim x->0 (x) * lim x->0 (x/2 * log(x))

The first limit, lim x->0 (x), is simply 0. For the second limit, we can use the fact that log(x) approaches negative infinity as x approaches 0. So, we have:

lim x->0 (x/2 * log(x)) = 0 * (-∞) = 0

Therefore, the limit of (x^2/2 * log(x)) as x approaches 0 is indeed 0.

2. To find the limit in equation (B), we need to evaluate the limit as x approaches infinity of [1/e^x * (-x^2-2x-2)].

As x approaches infinity, the exponential function e^x grows much faster than any polynomial function. Therefore, we can conclude that the term 1/e^x approaches 0 as x approaches infinity.

Now, we can rewrite the limit as:

lim x->infinity (0 * (-x^2-2x-2)) = 0

Therefore, the limit of [1/e^x * (-x^2-2x-2)] as x approaches infinity is indeed 0.

By evaluating these limits without using L'Hopital's Rule or the Squeeze Theorem, we have shown that it is possible to deduce the answers using simple algebraic simplification and the properties of the functions involved.