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**Nora314** Thank you so so much for the wonderful explanation! I went through it, and I believe I understand how to solve this exercise now. I just have a few questions, if you don't mind, which I think will help me clear things up completely.

What you do in the first part of the exercise, where you turn x^{2} + 1 / x + 1 into x - 1 + 2/x + 1, is it that you try to write the original expression in terms of x and not x^{2}, because we are trying to see if it is O(x)?

Also, when you write the whole new expression in terms for the largest positive value of x, and it becomes : 4|x|, why do you turn |2| / |x + 1| into 2|x| and not just |x|?