Question

Find each limit algebraically. $$ \lim _{x \rightarrow \infty} \frac{x-1}{3 x^{3}} $$

   Find each limit algebraically.
$$
\lim _{x \rightarrow \infty} \frac{x-1}{3 x^{3}}
$$

Precalculus

Raymond A. Barnett,… 7th Edition

Chapter 12, Problem 48

Instant Answer

Solved by Expert Christy Galilei

07/20/2021

Step 1

Step 1: First, we identify the highest power of x in the denominator, which is $x^3$. Show more…

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Find each limit algebraically. $$ \lim _{x \rightarrow \infty} \frac{x-1}{3 x^{3}} $$

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Key Concepts

Limits at Infinity

This concept involves evaluating the behavior of a function as the independent variable grows without bound. In the context of rational functions, limits at infinity help determine the end behavior of the function by comparing the degrees of the numerator and the denominator.

Degree Comparison in Rational Functions

When evaluating limits of rational functions at infinity, it is crucial to compare the highest power (degree) terms in the numerator and the denominator. If the degree of the numerator is lower than the degree of the denominator, the limit is typically zero, because the denominator grows much faster than the numerator.

Leading Term Analysis

This method involves isolating the dominant terms (those with the highest exponent) in both the numerator and the denominator to simplify the limit calculation. By focusing on the leading terms, one can often directly determine the limit as the negligible lower degree terms vanish when the variable becomes very large.

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