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, recognize that as \( x \to \infty \), \( \frac{1}{x} \to 0 \). Therefore, the expression inside the logarithm becomes \( \cos(\frac{1}{x}) \to \cos(0) = 1 \).
2. Substitute \( y = \frac{1}{x} \), so as \( x \to \infty \), \( y \to 0^+ \). The limit becomes \( \lim_{y \to 0^+} \frac{\ln(\cos(y))}{y^2} \).
3. Notice that both the numerator \( \ln(\cos(y)) \to \ln(1) = 0 \) and the denominator \( y^2 \to 0 \) as \( y \to 0^+ \). This is an indeterminate form \( \frac{0}{0} \), so l'Hôpital's Rule is applicable.
4. Apply l'Hôpital's Rule: Differentiate the numerator and the denominator with respect to \( y \). The derivative of \( \ln(\cos(y)) \) is \( -\tan(y) \) and the derivative of \( y^2 \) is \( 2y \).
5. Re-evaluate the limit: \( \lim_{y \to 0^+} \frac{-\tan(y)}{2y} \). Simplify and evaluate this new limit to find the solution.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches infinity helps determine the behavior of the function x² ln(cos(1/x)) at large values of x.

l'Hôpital's Rule

l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in simplifying complex limit problems, like the one presented.

Natural Logarithm and Cosine Function

The natural logarithm (ln) and the cosine function are key components in the limit expression. The cosine function approaches 1 as its argument approaches 0, which affects the behavior of ln(cos(1/x)). Understanding how ln behaves near 1 is crucial, as it approaches 0, influencing the overall limit when multiplied by x².