I understand very well that the latter conclusion is false. However, I can't explain the error in reasoning. I also don't get it what its meant by "taking the limit on both sides". Why does taking limit on both sides result in conclusion of the false value of integral?
Okay I understand what you are trying to say it, but can't prove it. You mention that the mistake is that interchanging limits and integrals won't generally work in unbounded intervals. Why wouldn't it work? is there any theorem or lemma out there, which can be used to explain this? I have searched my textbook countless times, and there is no mention of this.