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.

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$$f(t)=t(1-\frac{4}{t})t^3(1-\frac{2}{t})^3t^2(1-\frac{3}{t})^2$$f(t)=t^6(1-\frac{4}{t})(1-\frac{2}{t})^3(1-\frac{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.

$\lim_{t\to-\infty}g(t)=\operatorname{\lim}_{t\to-\infty}t^6$$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.