Evaluate the following limit:

lim_{x - 0} ln((x^2 + 1) cos(x)) / x^2

Question

Evaluate the following limit:

lim_{x -> 0} ln((x^2 + 1) cos(x)) / x^2

Accepted Answer

0.5 Using the logarithm property ln(ab) = ln(a) + ln(b), the limit splits into two parts: ln(x^2+1)/x^2 and ln(cos x)/x^2. Each can be evaluated using standard Taylor expansions: ln(1+u) ~ u for small u, and cos x ~ 1 - x^2/2, so ln(cos x) ~ -x^2/2. Combining gives 1 + (-1/2) = 1/2.