Answer to Question #95107 in Calculus for ana

Question

Determine the end (long run) behavior for the function below:

f(x)=x4−3x3+x−1

Answer 1

The a long run behaviour of polynomial function is determined by its leading term (with the highest power). It means, that for long run asymptotics function $f(x) = {x^4} - 3{x^3} + x - 1$ is equivalent to the simple function $g(x) = {x^4}$.

$g(x)$ is even function, so asymptotics for $f(x)$ in both directions are the same. To be more exact, for

$x \to + \infty$ we've got $f(x) \to + \infty$

and for

$x \to - \infty$ we've got $f(x) \to + \infty$ .

The graph of the given function $f(x)$ is drawn below