Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ x² ln( cos 1/x)

Verified step by step guidance

1. First, rewrite the limit as lim_{x→∞} x² ln(cos(1/x)). As x approaches infinity, 1/x approaches 0, so we need to analyze the behavior of ln(cos(1/x)) as x approaches infinity.
2. Next, note that cos(1/x) approaches cos(0) = 1 as x approaches infinity. Therefore, ln(cos(1/x)) approaches ln(1) = 0.
3. Now, we have an indeterminate form of the type ∞ * 0. To apply l'Hôpital's Rule, we can rewrite the limit in a suitable form. Rewrite it as lim_{x→∞} ln(cos(1/x)) / (1/x²).
4. Now, check if this new limit is in the form 0/0 or ∞/∞. As x approaches infinity, both the numerator and denominator approach 0, so we can apply l'Hôpital's Rule.
5. Differentiate the numerator and denominator separately: the derivative of ln(cos(1/x)) and the derivative of (1/x²), then evaluate the limit again.