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One-sided limits

Level FundamentalDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Approaching $a$ only from the left ( $x \to a^-$ ) or only from the right ( $x \to a^+$ ). The limit exists exactly when both one-sided limits exist and agree.

Formulas

\lim_{x\to a} f(x) = L \iff \lim_{x\to a^-} f(x) = L = \lim_{x\to a^+} f(x)

\lim_{x\to 0^-}\frac{|x|}{x} = -1, \qquad \lim_{x\to 0^+}\frac{|x|}{x} = 1

Where is it used?

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

λ Scientific computing and algorithms

Differentiability and one-sided derivatives→Rolle's theorem→Mean value theorem→Taylor's theorem and the remainder→Scientific computing★★★★★
Differentiability and one-sided derivatives→Rolle's theorem→Mean value theorem→Fundamental theorem of calculus→Symbolic computation (CAS)★★★★★
Differentiability and one-sided derivatives→Rolle's theorem→Mean value theorem→Taylor's theorem and the remainder→Floating point (IEEE 754)★★★★★

ℒ AI and machine learning

Differentiability and one-sided derivatives→Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent★★★★★
Discontinuities→Activation functions★★★★★
Discontinuities→Activation functions→Neural networks★★★★★
Discontinuities→Activation functions→Neural networks→Backpropagation★★★★★
Discontinuities→Activation functions→Neural networks→Loss landscape★★★★★
Discontinuities→Activation functions→Neural networks→Backpropagation→Deep learning★★★★★
+7

⚛ Physics and simulation

Differentiability and one-sided derivatives→Rolle's theorem→Mean value theorem→Fundamental theorem of calculus→Physics engines★★★★★

∿ Signals, media and vision

Discontinuities→Image processing and computer vision★★★★★
Discontinuities→Image processing and computer vision→Media compression (JPEG, MP3, video)★★★★★

What depends on it

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.