Question

Describe the long run behavior: f(t) = (t-4) (t-2)^3 (t-3)^2 A. infinityB. negative infinityC. zeroD. N/A

Answer 1

Given:

`f(t)=(t-4)(t-2)^3(t-3)^2`

Required:

We need to find the long run behavior.

Step-by-step explanation:

Recall that the long-run behavior of a polynomial function is determined by its leading term.

Consider the given polynomial function.

`f(t)=(t-4)(t-2)^3(t-3)^2`
`f(t)=t(1-(4)/(t))t^3(1-(2)/(t))^3t^2(1-(3)/(t))^2`
`f(t)=t^6(1-(4)/(t))(1-(2)/(t))^3(1-(3)/(t))^2`
`\text{ The leading term is }t^6`
`Let\text{ }g(t)=t^6.`

Take the limit as infinity.

`\lim_(t\to\infty)g(t)=\lim_(t\to\infty)t^6`
`g(t)\rightarrow\infty\text{ as t approches }\infty`

Take the limit as negative infinity.

`g(t)\rightarrow\infty\text{ as t approches -}\infty.`

Hence we get

`f(t)\rightarrow\infty\text{ as t }\rightarrow\text{ -}\infty\text{ and t}\rightarrow\infty`

Final answer:

The long run behavior is infinity.