Horizontal asymptotes are horizontal lines approaching ±∞ in function graphs. Unlike vertical asymptotes, horizontal asymptotes don't appear on the graph. A function has a horizontal asymptote if either limit approaches ±∞
Horizontal asymptotes show where rational function's graph goes off to sides. Unlike vertical asymptotes, horizontal asymptotes can be touched and crossed. Horizontal asymptotes indicate general behavior, vertical asymptotes show specific behavior
The derivative of 2x is 2 as f'(x) = 2 for linear functions. The derivative of kx is k, so [2x]' = 2. The derivative of a constant function is always 0
Integration involves summing infinite small areas to find exact values. Definite integrals produce area under curves with upper/lower limits. Indefinite integrals produce anti-derivative equations without limits
A function is differentiable if its derivative exists at every point in its domain. The derivative is found by taking the limit of the average rate of change as Δx approaches 0. The difference quotient measures the slope of a function between two points
0.999... represents the smallest number greater than all numbers in the sequence. Elementary proof uses Archimedean property and decimal arithmetic. Proof shows sequence has least upper bound equal to 1. Real numbers can be constructed using Dedekind cuts or Cauchy sequences