Question

Find each limit algebraically. $$ \lim _{x \rightarrow-\infty}\left(-2 x^{3}-7 x+1\right) $$

   Find each limit algebraically.
$$
\lim _{x \rightarrow-\infty}\left(-2 x^{3}-7 x+1\right)
$$

Precalculus

Raymond A. Barnett,… 7th Edition

Chapter 12, Problem 44

Instant Answer

Solved by Expert Christy Galilei

07/20/2021

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In this case, it is $-2x^3$. Show more…

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Find each limit algebraically. $$ \lim _{x \rightarrow-\infty}\left(-2 x^{3}-7 x+1\right) $$

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Key Concepts

Dominance of the Leading Term

In polynomial functions, the term with the highest degree dictates the behavior of the function as x tends to infinity or negative infinity. Lower order terms become negligible compared to the leading term, allowing us to determine the limit based primarily on this dominant part.

End Behavior of Polynomial Functions

The end behavior of a polynomial is determined by its highest power. For x approaching negative infinity, the sign and degree of the leading term indicate whether the function will approach positive or negative infinity. In this context, the sign of the leading coefficient is crucial in predicting how the function behaves at extreme values of x.

Limits at Infinity

When evaluating limits at infinity, one analyzes the function's behavior as the variable grows without bound. For polynomial functions, this involves focusing on the dominant term. This method simplifies the computation by ignoring lower order terms that do not affect the asymptotic outcome.

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