Question

What is the end behavior of function g?

Use limit notation to describe the end behavior of function g.

Answer 1

Read below.

Explanation:

When we are asked for end behaviors, we ask ourselves,

What is `lim_(x->oo)((-4x^3+4x^2-2x-4)/(x^2+8x+2))`?

What is `lim_(x->-oo)((-4x^3+4x^2-2x-4)/(x^2+8x+2))`?

We will use logic here.

As `x` gets really, really large/small, only the variable to its highest degree will matter.

For example, in `lim_(x->oo)x^2-x`, `x^2` is so large that `x` merely matters.

We could see this as `lim_(x->oo)x^2`, which we now see will be unboundedly large.

Therefore, we turn `(-4x^3+4x^2-2x-4)/(x^2+8x+2)` to `(-4x^3)/(x^2)`

We now simplify this.

`(-4x^3)/(x^2)=>-4x` Now, when `x` gets really, really large, we see that `y` will get unboundedly small. (We are multiplying `oo` by a negative number)

Similarly, as `x` gets really, really small, we see that `y` will get unboundedly large.(We are multiplying `-oo` by a negative number)

Therefore, `lim_(x->oo)((-4x^3+4x^2-2x-4)/(x^2+8x+2))=-oo`
and `lim_(x->-oo)((-4x^3+4x^2-2x-4)/(x^2+8x+2))=oo`