Describe in words the long run behavior as x approaches infinity of the function

y = 6x^6 + (4x^4/x^-9) -9x^7+3

I know y goes to positive infinity as x approaches infinity

The graph resembles ? (is it 4x^5)

I tried simplifying 4x^4/x^-9 and got 4/x^-5 is what the graph resembles is this right?

y = 6x^6 + 4x^13 - 9x^7 + 3

for large x, it's just 4x^13

To determine the long-run behavior of the function y = 6x^6 + (4x^4 / x^-9) - 9x^7 + 3 as x approaches infinity, we can analyze the dominant term(s) in the function.

Let's simplify the expression (4x^4 / x^-9) first. When dividing with negative exponents, we can rewrite the expression as (4x^4 * x^9). By applying the rule of exponents and multiplying the like terms, we get 4x^13.

So the original function can be written as y = 6x^6 + 4x^13 - 9x^7 + 3.

Now, as x approaches infinity, we focus on the terms that have the highest power of x. In this case, the dominant term is 4x^13.

As x becomes larger and larger, the term 4x^13 will grow at a much faster rate compared to the other terms. Consequently, the function y will also approach positive infinity.

Regarding the graph of the function, it should resemble the graph of 4x^13, as this is the dominant term. Thus, the shape of the graph will be mainly determined by that term, which is a steep upward curve. Keep in mind that the other terms may also contribute to the overall shape, but their influence diminishes significantly as x gets larger.

In conclusion, the long-run behavior of the function y = 6x^6 + (4x^4 / x^-9) - 9x^7 + 3 as x approaches infinity is that y goes to positive infinity. The graph of the function most closely resembles the graph of 4x^13.