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)

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First, rewrite the expression inside the limit: x³ (1/x - sin(1/x)) = x³ * (1/x) - x³ * sin(1/x). This simplifies to x² - x³ * sin(1/x).

Consider the limit of each term separately as x approaches infinity. Start with the first term: lim_{x→∞} x². As x approaches infinity, x² also approaches infinity.

Now, consider the second term: lim_{x→∞} x³ * sin(1/x). As x approaches infinity, 1/x approaches 0, and sin(1/x) approaches sin(0), which is 0. Therefore, the expression becomes x³ * 0, which is 0.

Combine the results of the two limits: The first term approaches infinity, and the second term approaches 0. Therefore, the overall limit is dominated by the first term.

Conclude that the limit of the original expression as x approaches infinity is infinity, since the x² term grows without bound while the x³ * sin(1/x) term approaches 0.

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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 at extreme values.

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 simplifies the process of finding limits in complex expressions.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its input approaches a certain value, often infinity. Understanding this concept is crucial for evaluating limits at infinity, as it helps identify dominant terms in expressions. In the given limit, analyzing the behavior of x³ and sin(1/x) as x approaches infinity is key to finding the limit's value.

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)