Assume that a function f(x) is a monotonic function with a horizontal asymptote a as x approaches positive infinity, lim f(x) = a. What can we say about the limits of the first and second deriviatives of f(x) as x approaches positive infinity?
I should note that I imposed the restriction of monotonicity to rule out oscillating functions like f(x)=sin(x^2)/x, which has a horizontal asymptote at 0, but the limit of the first derivative does not exist.
After a nice run, the answer has come to me so I will share it here.
Both the first and second derivative must limit to zero as x approaches infinity as long as the first and second derivatives exist where the assumption or existence rules out these oscillating type counter examples.
Let's start with the first derivative. WOLOG, let's restrict the derivative to be positive. If it limits to infinity, then f(x) must limit to infinity contradicting the assumption of a horizontal asymptote. If it limits to a positive value, then for some epsilon we can find an x large enough so that the function never falls below that limit minus epsilon. This gives us a lower positive bound for the derivative and so as x goes to infinity the function value is bounded below by the product of x and that positive lower bound, and this product limits to infinity. Again contradicting the existence of a horizontal asymptote.
Now turning to the second derivative. The second derivative cannot limit to any positive finite value or positive infinity because if it did the positive first derivative could not limit to zero contradicting the previous result. If the second derivative has a negative limit, then for any small epsilon we can find an x large enough that the second derivative's magnitude never falls below that limit minus infinity. With a lower bound on the magnitude of the second derivative established, we can take the finite positive value of the first derivative at that large value x and then keep increasing x until the positive slope at the original x minus the lower bound on the second derivative times the change is x is negative. This then contradicts the restriction that the function be monotonically increasing. Therefore, the second derivative must limit to zero.