Question

Find the limits. $$ \lim _{t \rightarrow-\infty} \frac{5-2 t^{3}}{t^{2}+1} $$

   Find the limits.
$$
\lim _{t \rightarrow-\infty} \frac{5-2 t^{3}}{t^{2}+1}
$$

Calculus Early Transcendentals

Calculus Early Transcendentals

Howard Anton, Irl… 9th Edition

Chapter 1, Problem 20

Instant Answer

Solved by Expert Amy Jiang

08/14/2020

Step 1

This is done by dividing each term in the numerator and the denominator by $t^{3}$, the highest power of t in the numerator. This gives us: $$ \lim _{t \rightarrow-\infty} \frac{\frac{5}{t^{2}}-2 t}{1+\frac{1}{t^{2}}} $$ Show more…

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Find the limits. $$ \lim _{t \rightarrow-\infty} \frac{5-2 t^{3}}{t^{2}+1} $$

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Key Concepts

Dominant Term Analysis

This technique focuses on identifying and isolating the terms with the highest degrees in both the numerator and the denominator. Because these terms grow much more rapidly than the lower degree terms as the variable approaches infinity, they effectively determine the value of the limit, allowing for a simplified analysis.

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. Analyzing their limits as the variable approaches infinity involves understanding how the highest degree terms in the numerator and denominator dictate the end behavior of the function.

Limits at Infinity

This concept involves evaluating the behavior of a function as the variable grows without bound in the positive or negative direction. For rational functions, the limit at infinity is often determined by comparing the degrees of the numerator and the denominator, which reveals whether the function approaches zero, a finite constant, or infinity.

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