what is the mistake here

Question

Answer 1

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lim x--> 0 (1/x^2) = infinity, and you can't write infinity - infinity = 0; i.e. that is false

Answer 2

As it is in 0/0 form so you can apply L'hopital Rule too here from there
You will get same answer as 1/2

Answer 3

Look carefully at the end you have 1/xsq which tends to inf as x tends to 0 and similarly 1/xsq tends to zero as x tends to inf so this is really infinity - infinity indeterminate form and "agar aisa answer aaye toh pure solution par kata ragiye or bina distribute kare solve kariye"(sachin jakhar sir)  

Your doubt is solved by this note. (which reads in english as If you distributed the limits in addition or subtraction, then while giving the final answer do NOT re-club the terms and calculate them INDEPENDENTLY."
Note 2 is also valid as you are getting indeterminate form after distributing the limits. 
the correct rule Note 4 says distribute only when both limits have finite value.