Determine the following limits. lim x→−∞ (5 + 100/x + sin4 x3 / x2)

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 20

Chapter 2, Problem 20

Determine the following limits.
lim x→−∞ (5 + 100/x + sin⁴ x³ / x²)

Verified step by step guidance

Identify the limit expression: $ \lim_{x \to -\infty} \left( 5 + \frac{100}{x} + \frac{\sin^4(x^3)}{x^2} \right) $.

Recognize that as $ x \to -\infty $, the term $ \frac{100}{x} $ approaches 0 because the numerator is constant and the denominator grows without bound.

Consider the term $ \frac{\sin^4(x^3)}{x^2} $: since $ \sin(x^3) $ is bounded between -1 and 1, $ \sin^4(x^3) $ is also bounded between 0 and 1.

As $ x \to -\infty $, the denominator $ x^2 $ grows without bound, making $ \frac{\sin^4(x^3)}{x^2} $ approach 0.

Combine the results: the limit simplifies to $ 5 + 0 + 0 = 5 $, so the limit is 5.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. In this context, we analyze how each term in the function behaves as x approaches negative infinity, which helps determine the overall limit.

Dominant Terms

In limit problems, dominant terms are those that have the most significant impact on the function's value as x approaches a certain point. For large values of x (positive or negative), terms with higher powers of x typically dominate, while lower power terms and constants become negligible.

Trigonometric Limits

Trigonometric limits involve understanding the behavior of trigonometric functions as their arguments approach certain values. In this case, the term sin^4(x^3) oscillates between 0 and 1, but its contribution diminishes when divided by x^2 as x approaches negative infinity.

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