What is strong function and weak function.

Question

One of our professors told us that in an expression like $xe^x$ or $x+e^x$ the function behaves like $e^x$ rather than $x$ in the long run i.e. for larger $x$. He told that this dominance is because $e^x$ is a strong function w.r.t. any linear function. Is there any formal definition or rule to determine which function will dominate in a general case?

Answer 1

Simply put, polynomials dominate logarithmic functions, exponentials dominate polynomials, and factoials dominate exponentials.

That is as $x$ gets very large, the absolute value of the ratio of the dominating function over the weaker one goes to infinity.

For example $$\lim _{x\to \infty } \frac {e^x}{x^2+3x+5} =\infty$$ $$\lim _{n\to \infty } \frac {n!}{3^n} =\infty$$