apex vs upper limit

apex

noun
  • The top of the food chain. 

  • The moment of greatest success, expansion, etc. 

  • The end or edge of a vein nearest the surface. 

  • A conical priest cap. 

  • The highest point in a plane or solid figure, relative to a base line or plane. 

  • The end of a leaf, petal or similar organ opposed to the end where it is attached to its support. 

  • The growing point of a shoot. 

  • The deepest part of a tooth's root. 

  • A diacritic in Middle Vietnamese that indicates /ŋ͡m/. 

  • The highest or the greatest part of something, especially forming a point. 

  • A diacritic in Classical Latin that resembles and gave rise to the acute. 

  • The lowest part of the human heart. 

  • The point on the celestial sphere toward which the Sun appears to move relative to nearby stars. 

  • A sharp upward point formed by two strokes that meet at an acute angle, as in "W", uppercase "A", and closed-top "4", or by a tapered stroke, as in lowercase "t". 

  • The lowest point on a pendant drop of a liquid. 

upper limit

noun
  • The upper limit of a sequence of real numbers is the real number which can be found as follows: remove the first term of the sequence in order to obtain the "first subsequence." Then remove the first term of the first subsequence in order to obtain the "second subsequence." Repeat the removal of first terms in order to obtain a "third subsequence," "fourth subsequence," etc. Find the supremum of each of these subsequences, then find the infimum of all of these supremums. This infimum is the upper limit. 

How often have the words apex and upper limit occurred in a corpus of books? (source: Google Ngram Viewer )