Question

Find the horizontal asymptote of the graph of y=-4x^6+6x+3/8x^6+6x+5?

Answer 1

`color(blue)(y=-1/2)`

Explanation:

I am assuming the function is:

`y=(-4x^6+6x+3)/(8x^6+6x+5)`

Divide numerator and denominator by `x^6`

`y=((-4x^6)/x^6+(6x)/x^6+3/x^6)/((8x^6)/x^6+(6x)/x^6+5/x^6)`

Cancelling:

`y=(-4+(6)/x^5+3/x^6)/(8+(6)/x^5+5/x^6)`

Now we observe what happens as `x->+-oo`

`x->oo` ,`\ \ \ \ \ \(-4+(6)/x^5+3/x^6)/(8+(6)/x^5+5/x^6)->(-4+0+0)/(8+0+0)=-1/2`

`x->-oo` ,`\ \ \ \ \ \(-4+(6)/x^5+3/x^6)/(8+(6)/x^5+5/x^6)->(-4+0+0)/(8+0+0)=-1/2`

We can see that as `x` tends to `+-` infinity the function tends to `-1/2`

The line `y=-1/2` is therefore a horizontal asymptote.

The graph of the function verifies this:

graph{y=(-4x^6+6x+3)/(8x^6+6x+5) [-10, 10, -5, 5]}