17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→∞ x³ (1/x - sin 1/x)

Verified step by step guidance

1. Rewrite the limit as lim_{x→∞} x³ (1/x - sin(1/x)) to clarify the expression.
2. Simplify the expression inside the limit: 1/x approaches 0 as x approaches infinity, so sin(1/x) approaches sin(0) = 0.
3. Recognize that as x approaches infinity, both 1/x and sin(1/x) approach 0, leading to an indeterminate form of type ∞ * 0.
4. Rearrange the limit to a form suitable for l'Hôpital's Rule: lim_{x→∞} (x³ (1/x - sin(1/x))) = lim_{x→∞} (x² (1 - x sin(1/x))).
5. Apply l'Hôpital's Rule if necessary, differentiating the numerator and denominator until the limit can be evaluated.