GATE 2021 AG – Question 12
For integer x>1, evaluate $\lim_{x\to\infty}[1/x^2+2/x^2+\cdots+(x-1)/x^2+1/x]$.
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Correct answer: (B) .5
Explanation
The final1/x equals x/x²,so the bracket is $\sum_{k=1}^xk/x^2=x(x+1)/(2x^2)=(1+1/x)/2$. Its limit is **.5,B**. The number of terms grows with x and cannot be ignored when taking the limit.