Question

describe the long run behavior of $f(n) = 5n^9 - n^5 - 5n^3 + 4$
as $n \to -\infty$, $f(n) \to \boldsymbol{?}$
as $n \to \infty$, $f(n) \to \boldsymbol{?}$
question help: video written example

Explanation:

Step1: Identify the leading term

The leading term of the polynomial \( f(n) = 5n^9 - n^5 - 5n^3 + 4 \) is \( 5n^9 \), since it has the highest degree (9).

Step2: Analyze the degree and leading coefficient

• The degree of the polynomial is 9, which is odd.
• The leading coefficient is 5, which is positive.

Step3: Determine the long - run behavior for \( n

ightarrow-\infty \)
For a polynomial with an odd degree, as \( n
ightarrow-\infty \), if the leading coefficient is positive, we have:
When \( n
ightarrow-\infty \), \( n^9
ightarrow-\infty \) (because for an odd power, a negative number raised to an odd power is negative). And since the leading coefficient is 5 (positive), \( 5n^9
ightarrow-\infty \) as \( n
ightarrow-\infty \). The other terms (\( -n^5,- 5n^3,4 \)) become negligible compared to the leading term as \( n
ightarrow\pm\infty \). So as \( n
ightarrow-\infty \), \( f(n)
ightarrow-\infty \).

Step4: Determine the long - run behavior for \( n

ightarrow\infty \)
For a polynomial with an odd degree, as \( n
ightarrow\infty \), if the leading coefficient is positive, we have:
When \( n
ightarrow\infty \), \( n^9
ightarrow\infty \) (because a positive number raised to an odd power is positive). And since the leading coefficient is 5 (positive), \( 5n^9
ightarrow\infty \) as \( n
ightarrow\infty \). The other terms (\( -n^5,-5n^3,4 \)) become negligible compared to the leading term as \( n
ightarrow\pm\infty \). So as \( n
ightarrow\infty \), \( f(n)
ightarrow\infty \).

Answer:

As \( n
ightarrow-\infty \), \( f(n)
ightarrow-\infty \); As \( n
ightarrow\infty \), \( f(n)
ightarrow\infty \)