Which of the following functions grows the slowest as x goes to infinity?
x^e
e^x
ex
they all grow the same rate.
I think it is c because it does not have any exponents.
yes, ex would be linear,
To determine which of the given functions grows the slowest as x goes to infinity, we need to compare their growth rates.
Let's analyze each function:
1. x^e: This function has a variable base (x) raised to a constant exponent (e, which is approximately 2.718). As x goes to infinity, this function will grow faster than any polynomial function, but slower than exponential functions.
2. e^x: This function has a constant base (e) raised to a variable exponent (x). As x goes to infinity, this function grows exponentially faster because the exponent is increasing.
3. ex: This function has a constant base (e) multiplied by a variable (x). As x goes to infinity, this function grows exponentially faster because the exponent is increasing. It is worth noting that ex is equivalent to e^x.
We can conclude that the function that grows the slowest as x goes to infinity is x^e (option a). This is because it grows slower than the exponential functions (options b and c) but faster than a constant or linear function. Therefore, the correct answer is option a: x^e.
The function that grows the slowest as x goes to infinity among the given options is e^x. This can be determined by considering the growth rates of the functions.
As x goes to infinity, all exponential functions with positive exponents (such as x^e, e^x, and ex) grow at an increasing rate. However, among these options, the function e^x grows the slowest.
This is because when a constant base (e in this case) is raised to an increasing power (x), the resulting function exhibits exponential growth. However, the exponential function e^x grows at a slower rate compared to functions with variable bases (such as x^e and ex), as the power (x) increases.
Therefore, the correct answer is e^x.