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Advanced Placement /AP Calculus AB: 2.1.3 The Formal Definition of a Limit

AP Calculus AB: 2.1.3 The Formal Definition of a Limit

Advanced Placement6 CardsCreated 7 days ago

The formal definition of a limit uses ε (epsilon) and δ (delta) to precisely describe how a function approaches a value. It states that for every ε > 0, there exists a δ > 0 such that when x is within δ of c (but not equal to c), f(x) is within ε of the limit L.

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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.

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Key Terms

Term
Definition

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.

note

  • 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.

  • R...

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

0.0025

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


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

- |f(x)−L|

Hover to peek or log in to view all

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

x = c

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

0.03

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AP Calculus AB: 2.1.3 The Formal Definition of a Limit
TermDefinition

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.

note

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

0.0025

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


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

- |f(x)−L|

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

x = c

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

0.03