GATE 2021 BT – Question 33

Correct answer: 0.24 to 0.26

Explanation

At $x=0$ both the numerator and denominator are $0$, so use the Taylor series $\sin ax=ax-\dfrac{(ax)^3}{6}+\cdots$.

**Numerator:**
$$x-\sin2x=x-2x+O(x^3)=-x+O(x^3).$$

**Denominator:**
$$x-\sin5x=x-5x+O(x^3)=-4x+O(x^3).$$

**Limit:**
$$\lim_{x\to0}\frac{-x}{-4x}=\frac14=\mathbf{0.25}.$$

(Check by L'Hôpital: the derivative ratio is $\dfrac{1-2\cos2x}{1-5\cos5x}\to\dfrac{-1}{-4}=\dfrac14$.)