GATE 2025 AG – Question 37
Evaluate $\lim_{x\to0}(x\cos x-\sin x)/(x^2\sin x)$.
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Correct answer: (A) −1/3
Explanation
Use $\cos x=1-x^2/2+O(x^4)$ and $\sin x=x-x^3/6+O(x^5)$. The numerator is $x-x^3/2-x+x^3/6+O(x^5)=-x^3/3+O(x^5)$,while the denominator is $x^3+O(x^5)$. Their ratio tends to **−1/3,A**.