Question: "So, I need to determine the end behavior..."

High School
Calculus and Analysis
Calculus 1
Limits and continuity

alka8q7

Answered question

2021-11-16

So, I need to determine the end behavior of P for this equation:

P (x) = - x^{5} + 3 x^{2} + x

If possible, how is this done?

Answer & Explanation

pseudoenergy34

Beginner2021-11-17Added 22 answers

Did you mean these?
$lim_{\to - \infty} P = + \infty$ and $lim_{+ \infty} P = - \infty$
This is basically because $x^{5}$ is the leading summ and it is taken with negative coefficient (-1).
For x of large absolute value, since $lim_{x \to \pm \infty} \frac{x^{k}}{x^{5}} = 0$ for $k < 5$ , so
$lim_{\pm \infty} P = lim_{x \to \pm \infty} - x^{5} (1 - \frac{3 x^{2} + x}{x^{5}}) = lim_{x \to \pm \infty} - x^{5}$

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