text: 10 terms
  
  # Calculus: Limits and Their Properties
  
  Intro to limits and their properties. (Symbols were not working when set created therefore old school/other means used) :)
  ###### PLAY
  Limit does not exist
  f(x) approaches different number from right side of c than it approaches from left side of c
  Limit does not exist
  f(x) increases or decreases without bound as x approaches c
  Limit does not exist
  f(x) oscillates between two fixed values as x approaches c
  Epsilon/Delta definition of a Limit
  Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then: lim f(x) = L (as x-->c) means that for each Epsilon>0 there exists a Delta>0 such that if 0<[x - c]<Delta, then [f(x) - L]<Epsilon.
  Derivative of a Function
  f'(x) = lim(as x-->0) {f(x+x) - f(x)}/*x provided the limit exists. For all x for which this limit exists, f' is a function of x.
  Intermediate Value Theorem (IVT)
  If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k
  f is continuous if
  f(c) is defined
  f is continuous if
  lim(as x-->c) f(x) exists
  f is continuous if
  lim(as x-->c) f(x) = f(c)
  Squeeze Theorem
  If h(x) <_f(x) <_ g(x) for all x in an open interval containing c, except possibly at c itself, and if lim(as x-->c) h(x) = L = lim(as x-->c) g(x) then lim(as x -->c) f(x) exists and is equal to L.
explanation: The document is a set of flashcards or study notes on calculus, specifically focusing on limits and their properties. It includes definitions, theorems, and conditions for the existence of limits and continuity. The content is presented in a question-and-answer format, where each term or concept is followed by its definition or explanation.
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distilabel_metadata.content: 10 terms
  
  # Calculus: Limits and Their Properties
  
  Intro to limits and their properties. (Symbols were not working when set created therefore old school/other means used) :)
  ###### PLAY
  Limit does not exist
  f(x) approaches different number from right side of c than it approaches from left side of c
  Limit does not exist
  f(x) increases or decreases without bound as x approaches c
  Limit does not exist
  f(x) oscillates between two fixed values as x approaches c
  Epsilon/Delta definition of a Limit
  Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then: lim f(x) = L (as x-->c) means that for each Epsilon>0 there exists a Delta>0 such that if 0<[x - c]<Delta, then [f(x) - L]<Epsilon.
  Derivative of a Function
  f'(x) = lim(as x-->0) {f(x+x) - f(x)}/*x provided the limit exists. For all x for which this limit exists, f' is a function of x.
  Intermediate Value Theorem (IVT)
  If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k
  f is continuous if
  f(c) is defined
  f is continuous if
  lim(as x-->c) f(x) exists
  f is continuous if
  lim(as x-->c) f(x) = f(c)
  Squeeze Theorem
  If h(x) <_f(x) <_ g(x) for all x in an open interval containing c, except possibly at c itself, and if lim(as x-->c) h(x) = L = lim(as x-->c) g(x) then lim(as x -->c) f(x) exists and is equal to L.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a set of flashcards or study notes on calculus, specifically focusing on limits and their properties. It includes definitions, theorems, and conditions for the existence of limits and continuity. The content is presented in a question-and-answer format, where each term or concept is followed by its definition or explanation.
  
  Label: multiple-question-and-answer