The Epsilon-Delta Definition of a Limit

A student-made presentation on the epsilon-delta definition of a limit.

1. The Epsilon-Delta Definition of a Limit A Calc I Topic

2. Find f ’(3) c) Prove your answer to part b b) Prove your answer to part a Exploration Let f(x) = 5x 2 f(x) = 5x 2 f’(x) = (5*2)x 2 -1 f’(x) = 10x f’(3) = 10(3) f’(3) = 30 Ummmm…?

3. Well… what is a limit? If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f ( x ), as x approaches c , is L , or lim f(x) = L x→c f(x) becomes “ arbitrarily ” close to L ? f(x) “ approaches ” c ? This seems pretty unspecific.

4. Breaking it down What if, instead, we use specific values to represent these vague terms? Take lim (2x - 3), for example. x →2 f(x) = 2x-3 becomes “arbitrarily” close to L ? How about f(x) is within .01 units of L ? This means that | f(x) - L | < .01. x “approaches” 2? Well, for all of those values of f(x) within .01 units of L , there is a corresponding value of x. And one of those values is 2, the x-value we’re approaching. This means that 0 < |x - 2 | < some value in relation to 0.01. ”If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L”

5. The Formal Definition of a Limit (The Epsilon-Delta Definition) Let f be a function defined on an open interval containing c (except possibly at c ) and let L be a real number. The statement l im f(x) = L x →c m eans that for every 𝜀 > 0, there exists a 𝛿 > 0 such that for every x , the expression 0 < | x - c | < 𝛿, implies | f(x) - L | < 𝜀.

6. In simpler terms, this means: if the distance between x and c is less than 𝛿, then the distance between the corresponding value of f(x) and the limit is less than 𝜀.

7. for every 𝜀 > 0 Our proof must work for every value of 𝜀 there exists a 𝛿 > 0 This is the key: we will need to give the value of 𝛿 to confirm its existence 0 < |x - c| < 𝛿 Our starting point for the proof, meaning that the distance between x and c will be less than 𝛿, and x will not be equal to c . implies |f(x) - L| < 𝜀 This is the conclusion. Once we reach this statement, the proof is complete. Breaking it down “for every 𝜀 > 0, there exists a 𝛿 > 0 such that for every x, the expression 0 < |x - c| < 𝛿, implies |f(x) - L| < 𝜀.”

8. Let’s Practice! ”for all 𝜀 > 0 there exists a 𝛿 > 0 such that if 0 < | x - c | < 𝛿, then | f(x) - L | < 𝜀” We will find 𝛿 by working backwards. First, we can sub in our known values, f(x) and L Then, we simplify with the goal of obtaining the form |x - c| < 𝛿 Since we were evaluating the limit as x approaches 2, the left sides of both of these inequalities are equal, so the right sides are equal, too. Given 𝜀 > 0 Choos e 𝛿 = Suppose 0 < |x - 2 | < 𝛿 Check | 2x-3 - 1 | 𝜀/2 ∴

9. Let’s Practice! ”for all 𝜀 > 0 there exists a 𝛿 > 0 such that if 0 < | x - c | < 𝛿, then | f(x) - L | < 𝜀” We start the same as last time, working backwards, trying to turn this into the form of |x - c| < 𝛿 We rewrite the absolute value this way because, since f(x) is non-linear, 𝛿 to the right side of x = c is likely not equal to 𝛿 to the left side of x = c Now that we’ve solved for x , we subtract 2 from all parts of the inequality to turn it into the form of |x - c| < 𝛿 Since there are two candidates for 𝛿, the left and right sides of the inequality, and 𝛿 must be less than or equal to both of them, we use the minimum of the two values as 𝛿. Given 𝜀 > 0 Choose 𝛿 = Suppose 0 < |x - 2 | < 𝛿 Check | 2x 2 -3 - 5 | ∴

10. Now you try! Given 𝜀 > 0 Choose 𝛿 = 𝜀/3 Suppose 0 < |x - 2 | < 𝛿 Check | 2x-3 - 1 | ∴

11. Application of the Epsilon-Delta Definition of a Limit At graduation, you throw your cap in the air to symbolize the end of this chapter of your life. The height of your cap above the ground in feet, h(x) , depends on the time in seconds after you threw it, t , and can be modeled by the function h(x) = -5x 2 + 12x. Using a limit, find the cap’s instantaneous velocity at t = 1 second. Prove your answer using an ϵ−δ proof.

12. Find f ’(3) c) Prove your answer to part b b) Prove your answer to part a Back to the Exploration Let f(x) = 5x 2 f(x) = 5x 2 f’(x) = (5*2)x 2-1 f’(x) = 10x f’(3) = 10(3) f’(3) = 30 Ummmm…? Given 𝜀 > 0 Choose 𝛿 = Suppose 0 < |x - 3 | < 𝛿 Check | (5x 2 - 5(3) 2 ) / (x-3) - 30 | < 𝜀

13. Works Cited “ The Hardest Calc 1 Topic, the Epsilon-Delta Definition of a Limit” . YouTube , uploaded by blackpenredpen, 15 Jan. 2022, https://www.youtube.com/watch?v=DdtEQk_DHQs. “Epsilon Delta Definition Of A Limit.” Calcworkshop , 22 Feb. 2021, https://calcworkshop.com/limits/epsilon-delta-definition/. Hartman, Gregory. “1.2: Epsilon-Delta Definition of a Limit.” Mathematics LibreTexts , NICE CXone Expert, 21 Dec. 2020, https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(Apex) /01%3A_Limits/1.02%3A_Epsilon-Delta_Definition_of_a_Limit. “How To Construct a Delta-Epsilon Proof.” How to Construct a Delta-Epsilon Proof , http://www.milefoot.com/math/calculus/limits/DeltaEpsilonProofs03.htm. LARSON, RON. “Limits and Their Properties.” Calculus of a Single Variable , 7th ed., CENGAGE LEARNING, S.l., 2022. “Limits and Continuity | AP®︎/College Calculus AB | Math.” Khan Academy , Khan Academy, https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new#ab-limits-optional.