2.1.3 The Formal Definition of a Limit

Question 1

The Formal Definition of a Limit

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

Given the limit
limx→2(2x+2)=6
find the largest value of δ such that ε<0.005.

Answer 3

0.0025

Question 4

The phrase “f (x) becomes arbitrarily close to L” means:

Answer 4

• f(x) lies in the interval (L−ε,L+ε).

- |f(x)−L|

Question 5

Which of the following is not an equivalent statement of  “x approaches c ?”

Answer 5

x = c

Question 6

Given the limit
limx→6(x/3+2)=4
find the largest value of δ such that ε<0.01.

Answer 6

0.03