Flashcards in 2.1.3 The Formal Definition of a Limit Deck (6):
The Formal Definition of a Limit
• The concept of a limit can be expressed exactly by describing it in terms of tiny neighborhoods that are mapped around a point.
• Formal Definition of a Limit Let f be a function defined on an open interval containing c(except possibly at c itself) and let L be a real number. If for each ε > 0 there exists a δ > 0 such that 0 < |x – c| < δimplies that |f(x) – L| < ε, then the limit as x approaches c exists and equals L.
- The idea of a limit as the value that you close in on from both directions can be easily be described in an intuitive way.
- Remember, the limit is the value that your fingers get infinitesimally close to when closing in on a particular x-value.
- One of the challenges of mathematics is to take these intuitive ideas and express them formally.
- The formal definition of a limit starts with a function defined on an open interval of radius δaround the x-value where you are taking the limit.
- If the limit exists, then every x-value in that interval is mapped to a y-value in another interval of radius ε that contains the limit.
- The trick is to show that shrinking one of the intervals shrinks the other interval. To do so, you must find a relationship between εand δ. If the limit exists, then there will be some sort of correlation between |x– c| and |f(x) – L|.
- Once you establish that relationship, then you have found the δ(in terms of ε) for which the limit holds. For example, if |f(x) – L| = 3|x– c|, then you can choose δ= ε/3. Thus, given any offset, you can select δsuch that the y-value is within that offset.
- At left is the formal definition of a limit.
Given the limit
find the largest value of δ such that ε<0.005.
The phrase “f (x) becomes arbitrarily close to L” means:
- f(x) lies in the interval (L−ε,L+ε).
Which of the following is not an equivalent statement of “x approaches c ?”
x = c