Question

How do you find the end behavior of `f(x) = x^6 + 2`?

Answer 1

Observe that the highest order term is `x^6`. This will be the dominant term for large values of `x`. Since its coefficient is positive and its degree is even `f(x)->+oo` as `x->+-oo`

Explanation:

For polynomials, the end behaviour is dictated by the highest order term.

If the coefficient of the highest order term is positive and the degree is even then `f(x)->+oo` as `x->+-oo`.

If the coefficient of the highest order term is positive and the degree is odd then `f(x)->+oo` as `x->+oo` and `f(x)->-oo` as `x->-oo`.