Does x^4 * e^x have a horizontal asymptote?

Take the derivative to see if slope is zero.

dU/dx = x^4 * e^x
= 4x^3 * e^x + x^4 * e^x
So, what happens as x approaches neg large ?

To determine if a function has a horizontal asymptote, we need to evaluate what happens as x approaches positive and negative infinity separately.

As x approaches negative infinity, we need to observe the behavior of the function.

Taking the derivative of the function, we obtain dU/dx = 4x^3 * e^x + x^4 * e^x.

Now, as x approaches negative infinity, both x^3 and x^4 become very large negative numbers, while e^x approaches zero.

Since the exponential term, e^x, becomes small or approaches zero, and the polynomial terms, x^3 and x^4, become large negative numbers, the overall value of the function will become very small or approach zero.

Therefore, as x approaches negative infinity, the value of the function x^4 * e^x approaches zero.

To summarize, the function x^4 * e^x does have a horizontal asymptote at y = 0 as x approaches negative infinity.