IB DP Maths: AA HL复习笔记5.7.1 Basic Limits & Continuity

What are limits in mathematics?

• When we consider a limit in mathematics we look at the tendency of a mathematical process as it approaches, but never quite reaches, an ‘end point’ of some sort
• We use a special limit notation to indicate this

How do I find a simple limit?

How do I find a limit to infinity?

• If you attempt to evaluate a limit and get one of these two forms, you will need to try another strategy
• This may just mean different or additional algebraic rearrangement
• But it may also mean that you need to consider using l’Hôpital’s Rule or Maclaurin series to evaluate the limit

Do limits ever have ‘directions’?

• Yes they do!

What does it mean for a function to be continuous at a point?

• If a function is continuous at a point then the graph of the function does not have any ‘holes’ or any sudden ‘leaps’ or ‘jumps’ at that point
  • One way to think about this is to imagine sketching the graph
    • So long as you can sketch the graph without lifting your pencil from the paper, then the function is continuous at all the points that your sketch goes through
    • But if you would have to lift your pencil off the paper at some point and continue drawing the graph from another point, then the function is not continuous at any such points where the function ‘jumps’

• There are two main ways a function can fail to be continuous at a point:
  • If the function is not defined for a particular value of x then it is not continuous at that value of x

What does it mean for a function to be differentiable at a point?

• We say that a function f(x) is differentiable at a point with x-coordinate x₀, if the derivative f’(x) exists and has a well-defined value f’(x₀) at that point

• To be differentiable at a point a function has to be continuous at that point

• • So if a function is not continuous at a point, then it is also not differentiable at that point

• But continuity by itself does not guarantee differentiability

• • This means that differentiability is a stronger condition than continuity
  • If a function is differentiable at a point, then the function is also continuous at that point
  • But a function may be continuous at a point without being differentiable at that point
  • This means there are functions that are continuous everywhere but are not differentiable everywhere
• In addition to being continuous a point, differentiability also requires that the function be smooth at that point
  • ‘Smooth’ means that the graph of the function does not have any ‘corners’ or sudden changes of direction at the point
  • An obvious example of a function that is not smooth at certain points is a modulus function |f(x)| at any values of x where f(x) changes sign from positive to negative
    • At any such point a modulus function will not be differentiable