Evaluate the following limit: \(\lim_{{x \to 0}} \frac{{\ln((x^2 + 1) \cos…
2024
Evaluate the following limit:
\(\lim_{{x \to 0}} \frac{{\ln((x^2 + 1) \cos x)}}{{x^2}} \)
Attempted by 2 students.
Show answer & explanation
Correct answer: 0.5
Key idea: expand the logarithm and use Taylor expansions up to order x^2.
Write the logarithm as a sum: ln((1 + x^2) cos x) = ln(1 + x^2) + ln(cos x).
Use Taylor expansions for small x: ln(1 + x^2) = x^2 + o(x^2), and cos x = 1 - x^2/2 + o(x^2), so ln(cos x) = ln(1 - x^2/2 + o(x^2)) = -x^2/2 + o(x^2).
Add the two expansions: ln(1 + x^2) + ln(cos x) = x^2 + (-x^2/2) + o(x^2) = x^2/2 + o(x^2).
Divide by x^2 and take the limit: (x^2/2 + o(x^2))/x^2 -> 1/2.
Answer: 1/2
Note: An alternative method is to apply l'Hôpital's rule twice to the original 0/0 form, which leads to the same result 1/2.