Intro to limits

• Limit


  The limit of a sequence or function as the variable approaches a certain value
• Left-hand limit

  The limit of the function as x approaches a from the left (x < a)
• Right-hand limit


  The limit of the function as x approaches a from the right (x > a)
• A limit exists if and only if the left and right limits both exist and are equal
• If the left and right limits are not equal, then the limit does not exist
• The function does not have to be defined at the point x = a for the limit to exist
• The limit must be a number, ∞ is not a number
• Direct substitution only works if the function is continuous at that x-value
• Continuous function

  A function y = f(x) is continuous at x = a if the limit of f(x) as x approaches a equals f(a)
• For a function to be continuous at x = a, 3 conditions must be met:

  1. f(a) must exist
  2. The two-sided limit must exist at x = a
  3. The value of the function and the limit must be equal
• Types of discontinuities

  • Break due to hole or displaced point but one-sided limits are the same value
  • Break where one-sided limits at a point have different values
  • Break due to vertical asymptote