In mathematics, a negligible function is a function $\mu(x):\mathbb{N}{\rightarrow}\mathbb{R}$ such that for every positive integer $c$ there exists an integer $N_c$ such that for all $x > N_c$, $\mu(x)<\frac{1}{x^c}$. (From wikipedia)
What exactly is $N_c$?
The definition is quite confusing for me, but I think the intuition is that a negligible function is something that shrinks faster than 1 divided by any polynomial function?
Is $2^{-x}$ negligible?
I think so, but don't know how to show this.
Is $x^{-100}$ negligible?
No, not negligible. The condition can be written as $x^{-100} < x^{-c}$ but this will only hold for values of $c$ that are smaller than $100$
Is $2^x$ negligible?
No, this is clear because as $\lim_{x->\infty} 2^x = \infty$, whereas $\lim_{x->\infty}x^{-c} = 0$, thus the inequality will not hold