Limits at infinity

What is it?

Formulas

$$\lim_{x\to\infty}\frac{1}{1 + e^{-x}} = 1, \qquad \lim_{x\to\infty}\frac{3x^2 + x}{x^2 - 5} = 3$$

Where it shows up in computing

• Algorithm analysis and complexity★★★★★fundamentalScientific computing and algorithms

  $f = \Theta(g)$ when $f(n)/g(n)$ stays between positive constants as $n \to \infty$.

Where it shows up in AI

• Activation functions★★★★★frequentAI and machine learning

  The sigmoid saturates: as $|x| \to \infty$ its slope tends to 0, the root of vanishing gradients.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Orders of growth
• Improper integrals