How do you find the end behavior of y = 2+3x-2x^2-x^3?

May 3, 2015

The end behavior of a function is the behavior of the function as x approaches positive infinity or negative infinity.

So we have to do these two limits:

${\lim}_{x \rightarrow - \infty} f \left(x\right)$

and

${\lim}_{x \rightarrow + \infty} f \left(x\right)$.

Than:

${\lim}_{x \rightarrow - \infty} \left(2 + 3 x - 2 {x}^{2} - {x}^{3}\right) = {\lim}_{x \rightarrow - \infty} \left(- {x}^{3}\right) = + \infty$

and

${\lim}_{x \rightarrow + \infty} \left(2 + 3 x - 2 {x}^{2} - {x}^{3}\right) = {\lim}_{x \rightarrow + \infty} \left(- {x}^{3}\right) = - \infty$.

This is because the power $- {x}^{3}$ is the highest power and its behavior is the same of the whole function.