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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 18
Chapter 2, Problem 18

Determine the following limits. 
lim t→∞ (5t2 + t sin t) / t2

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Identify the dominant terms in the numerator and the denominator. In this case, the dominant term in the numerator is \$5t^2\( and in the denominator is \)t^2$.
Divide every term in the numerator and the denominator by \(t^2\) to simplify the expression.
The expression becomes \(\frac{5t^2}{t^2} + \frac{t \sin t}{t^2}\) in the numerator and \(\frac{t^2}{t^2}\) in the denominator.
Simplify the expression to \(5 + \frac{\sin t}{t}\).
Evaluate the limit as \(t \to \infty\). The term \(\frac{\sin t}{t}\) approaches 0 as \(t\) becomes very large, so the limit is determined by the constant term 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 variable approaches infinity. This concept is crucial for understanding how functions behave for very large values, which often simplifies the analysis of rational functions and helps determine horizontal asymptotes.
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Dominant Terms

In the context of limits, dominant terms are the terms in a polynomial or rational function that have the greatest impact on the function's value as the variable approaches infinity. Identifying these terms allows for simplification of the limit, as lower-order terms become negligible compared to the dominant ones.
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L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. By differentiating the numerator and denominator separately, this rule can simplify the limit calculation, making it easier to find the limit's value as the variable approaches a specific point or infinity.
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Related Practice
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