You already know how to analyze the behavior of a ratio with fixed numerator, by considering what's happening in the denominator. That is, as the denominator grows without bound, while the numerator is fixed, the value of the ratio heads toward zero.
Are you familiar with the behavior of e^x, for all Real numbers?
If so, you could focus on the e^(3x) term, and work through the following limit analyses, step-by-step.
What happens to the value of e^(3x), as x increases without bound? How does this affect the entire denominator? Hence, what is the resulting behavior of the ratio f(x), as x heads toward positive infinity?
What happens to the value of e^(3x), as x becomes increasingly negative? What effect does this have on the denominator, and, therefore, on the value of f(x)?
What's the value of f(0)? What happens to function f, as x starts to move away from zero, in either direction?
Your conclusions ought to give you a good idea of function f's global behavior, and that's enough to answer both parts.
If you know how to determine the first derivative of f(x), you could also analyze that as well, to discover that f'(x) is always negative. Do you know: Functions increase, when their first derivative is positive; they decrease, when the first derivative is negative.
Questions about this approach? :cool: