# Continuous Function Outgrowing Polynomial but Not Exponential

• disregardthat
In summary, there exists a continuous function that outgrows polynomial growth but not exponential growth. This can be demonstrated by taking the logs of the polynomial and exponential functions and finding a function that grows faster than the log function but slower than the exponential function. Examples of such functions include x^{\sqrt{x}}, e^{\sqrt{x}}, and e^{x/\ln{x}}.
disregardthat
Does there exist a continuous function which outgrows polynomial growth, but not exponential growth?

I.e. does a there exist a continuous function f such that $$\frac{x^n}{f(x)} \to 0$$ and $$\frac{f(x)}{a^x} \to 0$$ for all positive real n and a?

Yes. Look at it this way: you take logs of the polynomial and the exponential, you get $$g_1(x) = C_1 \ln x$$ and $$g_2(x) = C_2 x$$. Can you find a function that grows faster than g_1 and slower than g_2 for all C? Clearly you can, because ln grows extremely slowly.

Thanks,

$$x^{\sqrt{x}}$$ is such a function.

Or $$e^{\sqrt{x}}$$, or $$e^{x/\ln{x}}$$.

I'm sure we can find many as you pointed out.

## 1. What is a continuous function?

A continuous function is a mathematical concept where the graph of the function has no breaks or gaps. This means that the output values of the function change smoothly and continuously as the input values change.

## 2. How does a function outgrow a polynomial but not an exponential?

A polynomial function grows at a slower rate than an exponential function. This means that for large input values, the output of an exponential function will be significantly larger than that of a polynomial function. However, a continuous function can have a graph that is curved in such a way that it outgrows a polynomial function without reaching the rapid growth of an exponential function.

## 3. Can a continuous function outgrow a polynomial function without being an exponential function?

Yes, it is possible for a continuous function to outgrow a polynomial function without being an exponential function. This occurs when the graph of the function has a curved shape that allows it to have a higher growth rate than a polynomial function, but not as high as an exponential function.

## 4. What is an example of a continuous function outgrowing a polynomial but not an exponential?

An example of a continuous function that outgrows a polynomial function but not an exponential function is the function f(x) = x^3 + 2x. This function has a curved graph that outgrows a polynomial function (x^3) but does not have the rapid growth of an exponential function.

## 5. Why is it important to understand continuous functions that outgrow polynomials but not exponentials?

Understanding this concept is important in mathematics and science because it allows us to model and analyze real-world phenomena that have continuous growth. By using continuous functions, we can make more accurate predictions and understand the behavior of various systems and processes. Additionally, it helps us to differentiate between different types of growth and better understand the limitations of polynomial and exponential functions.

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