GATE 2024 DA – Question 60
Evaluate the following limit: $\lim_{x \to 0}\frac{\ln((x^2 + 1)\cos x)}{x^2} = $ ______.
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Correct answer: 0.49 to 0.51
Explanation
Split the logarithm: $\frac{\ln(1 + x^2)}{x^2} + \frac{\ln\cos x}{x^2}$. The first term tends to 1. For the second, $\ln\cos x \approx \ln\left(1 - \frac{x^2}{2}\right) \approx -\frac{x^2}{2}$, so it tends to $-\frac{1}{2}$. The limit is $1 - \frac{1}{2} = 0.5$.