# Bounding e raised to a polynomial - Tell me if this is true?

• Skatch
In summary, the statement is true and can be proven by choosing the appropriate values for A and B in order to ensure that the right hand side always dominates the left hand side. This can be done by using the fact that |P(x)| and |Q(x)| can be bounded by exponential functions.
Skatch
Can someone tell me if the following statement is true?

Say you have $$P(x)e^{Q(x)}$$ where P(x) is any finite-degree polynomial and Q(x) is a polynomial of integer order k. Is it true that for any positive epsilon, I can find some real numbers A and B such that

$$|P(x)e^{Q(x)}| \leq Ae^{B|x|^{k+\epsilon}}\quad \forall x ?$$

I don't need a proof, just some intuition about why its true or a counterexample if its not. I figure since the expression on the right will be growing faster than that on the left, I should be able to find some A and B to ensure the RHS will always be larger, right? Is this an obvious statement or am I crazy??

Edit: I think its true since the smallest the right hand side can be is A at x=0. So I choose A so that the RHS is larger at x=0, and B is chosen to ensure that its larger around zero, and then far from zero the growth of the exponent on the RHS will dominate and we are home free... This is the idea, yes?

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Intuition:
|P(x)| < Ce|x|ε for some C and ε > 0. Similarly |Q(x)| < A|x|k for some A. Put it all together to get what you want.

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## 1. Is e raised to a polynomial always bounded?

Yes, e raised to a polynomial is always bounded. This is because the exponential function has a limit of infinity as the input approaches infinity, but a polynomial has a finite limit as the input approaches infinity. Therefore, the combination of the two functions results in a bounded output.

## 2. What is the significance of e in this context?

The number e, also known as Euler's number, is a mathematical constant that is the base of the natural logarithm. It is an important number in many mathematical and scientific contexts, including the exponential function, which is used to describe growth and decay processes. In the context of a polynomial, e helps to keep the output bounded.

## 3. Can a polynomial be raised to e?

Yes, a polynomial can be raised to e. This would result in a function that is not bounded, as the exponential function would dominate the polynomial function. However, if the polynomial is raised to a negative power of e, the output would be bounded.

## 4. How can we prove that e raised to a polynomial is bounded?

One way to prove that e raised to a polynomial is bounded is to use the limit definition of a bounded function. By taking the limit of e raised to the polynomial as the input approaches infinity, we can see that the output will approach a finite limit, thus proving that the function is bounded.

## 5. Can a polynomial with a negative exponent be bounded when raised to e?

No, a polynomial with a negative exponent cannot be bounded when raised to e. This is because the negative exponent will result in a fraction, which will have a limit of infinity as the input approaches infinity. Therefore, the combination of the polynomial and e would result in an unbounded output.

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