Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy

1. Diagram
   Observation

   A horizontal blue line is drawn and labeled as the x-axis.

x

Independent variable axis

Real numbers

1. Diagram
   Observation

   A vertical gray line is drawn and labeled as the y-axis.

y

Dependent variable axis

Real numbers

1. Audio
   Observation

   The speaker says 'let's say that this is f of x'.

2. Diagram
   Observation

   A yellow line with a hole is drawn and labeled $y=f(x)$.

$f(x)$

Function value at x

Real numbers excluding a

1. Audio
   Observation

   The speaker says 'except it has a hole at some point x is equal to a'.

2. Diagram
   Observation

   A green tick mark labeled 'a' is placed on the x-axis below the hole.

a

Specific x-value where the function is undefined

Real number

1. Audio
   Observation

   The speaker says 'let's say that this point right here is L'.

2. Diagram
   Observation

   A green dashed line extends from the hole to the y-axis, ending at a point labeled 'L'.

L

Limit value of the function as x approaches a

Real number

1. Formula
   Observation

   The expression $f(x)$ is written on the board and referenced in speech.

$f(x)$

function value at x

real numbers

1. Formula
   Observation

   The variable $x$ is written on the board as part of $x \to a$ and on the horizontal axis.

x

input variable approaching a

real numbers

1. Formula
   Observation

   The constant $a$ is written on the board as part of $x \to a$ and marked on the horizontal axis.

a

point that x approaches

real numbers

1. Formula
   Observation

   The constant $L$ is written on the board as part of $\lim_{x \to a} f(x) = L$ and marked on the vertical axis.

L

limit value of $f(x)$ as x approaches a

real numbers

1. Audio
   Observation

   The speaker says, 'let's call that epsilon'.

2. Formula
   Observation

   The symbol $\epsilon$ is written on the board next to the vertical distance markers.

$\epsilon$

arbitrary positive real number representing the allowed distance from L

real numbers greater than 0

1. Audio
   Observation

   The speaker says, 'I'll call that delta'.

2. Formula
   Observation

   The symbol $\delta$ is written on the board next to the horizontal distance markers.

$\delta$

distance around a that guarantees $f(x)$ is within epsilon of L

real numbers greater than 0

1. Diagram
   Observation

   Horizontal axis labeled 'x'.

x

Independent variable / input to the function.

Real numbers (implied by context).

1. Audio
   Observation

   The speaker explains the intuitive concept of a limit by describing approaching a point from both sides and seeing what value $f(x)$ approaches.

2. Diagram
   Observation

   Red arrows are drawn on the graph showing x approaching 'a' from the left and right, and the corresponding y-values approaching 'L'.

The limit of $f(x)$ as x approaches a is L if, as x gets closer to a from both the left and right sides, the value of $f(x)$ gets closer to L.

1. x must approach a from both sides

2. The left-hand and right-hand limits must be equal

1. Audio
   Observation

   The speaker states that the previous explanation was not rigorous and introduces the idea of defining a limit with more mathematical rigor, comparing it to a game.

1. The full formal definition involving epsilon and delta is not completed within this clip.

To make the concept of a limit mathematically rigorous, a more precise definition is needed beyond just saying 'gets closer'. This involves a game-like structure where one can always provide a specific range around the limit value.

1. Intuitive Definition of a Limit

1. Audio
   Observation

   The speaker explains the epsilon-delta definition of limits.

2. Formula
   Observation

   The expression $\lim_{x \to a} f(x) = L$ is written on the board.

The limit of $f(x)$ as $x$ approaches $a$ is $L$ if for any real number $\epsilon > 0$, there exists a real number $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

1. $\epsilon > 0$

2. $\delta > 0$

1. Audio
   Observation

   Speaker explains the relationship between choosing an epsilon (closeness to L) and finding a corresponding delta (closeness to a).

2. Diagram
   Observation

   Visual representation of epsilon intervals on the y-axis and delta intervals on the x-axis, showing how x-values within the delta interval map to $f(x)$-values within the epsilon interval.

The epsilon-delta definition formalizes the intuitive idea of a limit. For any desired closeness (epsilon) to the limit value L, there exists a corresponding closeness (delta) to the input value a such that if x is within delta of a (but not equal to a), then $f(x)$ will be within epsilon of L. The speaker describes this as a 'game' where one person picks an epsilon, and the other must find a working delta.

1. Applies to functions of a real variable.

2. Used to rigorously define the concept of a limit.

1. Formula
   Observation

   The mathematical notation for a limit is written on the screen.

2. Audio
   Observation

   Speaker states 'this is what you'll actually see in your math textbook' referring to the formal definition associated with this notation.

The standard way to express that the limit of a function $f(x)$ as x approaches a value 'a' is 'L' is using the limit notation.

1. a and L are real numbers.

2. f is a function defined on some open interval containing a, except possibly at a itself.

1. Formula
   Observation

   Written as 'give $ε > 0$' and 'They'll give a δ'.

2. Audio
   Observation

   Explained as a game where one person gives an epsilon and the other provides a delta.

The formal definition of a limit states that for any given positive number ε (epsilon), there exists a positive number δ (delta) such that if the distance between x and a is greater than zero and less than δ, then the distance between $f(x)$ and L is less than ε.

1. ε must be strictly greater than 0

2. δ must be strictly greater than 0

3. x must not equal a (0 < |x - a|)

1. Formula
   Observation

   Written as '|x - a|' and '|$f(x) - L$|'.

2. Audio
   Observation

   Described as 'the distance between x and a' and 'the distance between $f(x)$ and the limit point'.

The absolute value of the difference between two numbers represents the distance between them on the number line.

1. Formula
   Observation

   The board shows $\lim _{x\to a} f(x)=L$ in red at the upper right.

2. Audio
   Observation

   The speaker says that as x approaches this value, $f(x)$ is going to approach this value.

The displayed notation $\lim_{x\to a} f(x)=L$ states that when the input $x$ gets arbitrarily close to $a$, the corresponding output $f(x)$ gets arbitrarily close to $L$. In the clip, this symbolic statement is paired with a graph where $a$ lies on the horizontal axis and $L$ lies on the vertical axis.

1. a is the input value being approached

2. L is the proposed limiting output value

1. Formula
   Observation

   The lower-left board text reads 0<|x-a|<δ ⇒ |$f(x)-L$|<ε.

2. Audio
   Observation

   The speaker explains that if someone wants the distance between $f(x)$ and L to be very small, then one can always give a distance around x where this will be true.

The clip explicitly displays the core implication of the epsilon-delta definition: if $x$ is within $\delta$ of $a$ but not equal to $a$, then $f(x)$ is within $\epsilon$ of $L$. The left side controls input closeness to $a$; the right side controls output closeness to $L$.

1. $ε>0$ is given first

2. δ is then supplied in response

3. the implication is stated for x satisfying 0<|x-a|<δ

1. Limit notation

1. Formula
   Observation

   Green handwritten text reads give $ε>0$ and They'll give a δ.

2. Audio
   Observation

   The speaker says, 'you say ... I want the distance to be f of x and L ... point 0000001. Then I can always give you a distance around x where this will be true.'

1. The exact spoken numeral after 'point' is not fully clear, but it is presented as an extremely small positive tolerance.

The board and narration present the definition as a two-step process. First, a positive tolerance $\epsilon$ is chosen for the output distance from $L$. Second, a corresponding positive radius $\delta$ is provided for the input distance from $a$ so that the implication holds.

1. ε is chosen first

2. δ depends on the chosen ε

3. the goal is to make |$f(x)-L$| smaller than the prescribed ε

1. Displayed epsilon-delta implication

1. Audio
   Observation

   The speaker repeatedly describes the setup in terms of distance: 'I want the distance to be f of x and L' and 'a distance around x'.

2. Formula
   Observation

   The inequalities use absolute values |x-a| and |$f(x)-L$|.

In this clip, $|x-a|$ is used to mean the distance from $x$ to $a$ on the input axis, and $|f(x)-L|$ is used to mean the distance from $f(x)$ to $L$ on the output axis. The verbal explanation matches the algebraic form of the displayed inequalities.

1. distance is nonnegative

2. strict inequality < means 'within but not at the boundary'

1. Displayed epsilon-delta implication

1. Audio
   Observation

   The speaker says, 'This does make a lot of sense intuitively,' then explains the closeness idea verbally while pointing among the graph, the δ-interval, and the inequalities.

The displayed condition expresses the intuitive idea that making $x$ sufficiently close to $a$ forces $f(x)$ to be correspondingly close to $L$.

1. The graph shows a function approaching height L near input a

2. $ε>0$ is chosen as an output tolerance

3. δ is chosen as an input radius

For a given positive ε, there is a corresponding positive δ such that whenever 0<|x-a|<δ, the inequality |$f(x)-L$|<ε holds.

1. Diagram
   Observation

   A graph with axes, a curve, a vertical dashed line at a, a horizontal dashed line at L, an ε-band around L, and a δ-interval around a is visible throughout.

2. Formula
   Observation

   The board simultaneously shows $\lim _{x\to a} f(x)=L$ and 0<|x-a|<δ ⇒ |$f(x)-L$|<ε.

3. Audio
   Observation

   The speaker connects the visual neighborhoods to the written inequalities by describing closeness in words.

1. Expression

   $$\text{Graph: } x \text{ near } a,\ f(x) \text{ near } L$$
   Explanation

   The picture first shows the limiting behavior geometrically: inputs near $a$ produce outputs near $L$.

   Justification

   Directly visible in the coordinate diagram and reinforced by the spoken phrase about x approaching one value and $f(x)$ approaching another.

   Shown in the video
2. Expression

   $$\varepsilon>0 \text{ chosen on the output side}$$
   Explanation

   The horizontal band around $L$ represents a prescribed output tolerance $\epsilon$.

   Justification

   The board labels the vertical half-width as ε and writes 'give $ε>0$'.

   Shown in the video
3. Expression

   $$\delta>0 \text{ chosen on the input side}$$
   Explanation

   The interval around $a$ represents a corresponding input radius $\delta$.

   Justification

   The board labels the horizontal half-width as δ and writes 'They'll give a δ'.

   Shown in the video
4. Expression

   $$0<|x-a|<\delta \Rightarrow |f(x)-L|<\varepsilon$$
   Explanation

   The formal implication states that any x inside the punctured δ-neighborhood of a must send $f(x)$ inside the ε-neighborhood of L.

   Justification

   This is exactly the formula written at lower left and is the algebraic translation of the pictured neighborhoods.

   Shown in the video

The clip presents the epsilon-delta definition as the formalization of the geometric idea that a small enough input neighborhood around $a$ forces the graph into a prescribed output neighborhood around $L$.

1. Audio
   Observation

   Speaker proposes a concrete example: 'Let's say you say I want $f(x)$ to be within 0.5... Let's say this is the number 2 and let's say this is number 1.'

2. Diagram
   Observation

   The speaker writes '0.5' next to the epsilon label, '2' next to the L label, and '1' next to the a label. He then draws new dashed lines corresponding to these values and labels the resulting delta interval with '0.1'.

Given a function where the limit as x approaches 1 is 2, demonstrate the epsilon-delta relationship with specific numbers.

1. Limit $L = 2$

2. Approach point $a = 1$

3. Chosen epsilon = 0.5

Find a corresponding delta that satisfies the condition.

1. Expression

   $$|f(x) - 2| < 0.5$$
   Explanation

   We want the function's output to be within 0.5 of the limit 2. This means $f(x)$ must be between 1.5 and 2.5.

   Justification

   Definition of absolute value inequality.

   Shown in the video
2. Expression

   $$0.9 < x < 1.1$$
   Explanation

   By looking at the graph (which appears to be a line with a hole, likely $y=x+1$), the speaker identifies an x-interval around 1 that maps to the y-interval (1.5, 2.5).

   Justification

   Visual inspection of the provided graph.

   Shown in the video
3. Expression

   $$\delta = 0.1$$
   Explanation

   The distance from the center point $a=1$ to either end of the interval (0.9 or 1.1) is 0.1. So, a delta of 0.1 works.

   Justification

   Calculation of distance on the number line.

   Shown in the video

For epsilon = 0.5, a valid delta is 0.1.

If x is within 0.1 of 1 (i.e., $0.9 < x < 1.1$), then based on the visual graph, $f(x)$ will be within 0.5 of 2 (i.e., $1.5 < f(x) < 2.5$).

1. Diagram
   Observation

   Axes are drawn, followed by a linear function with a hole at $x=a$, and labels for axes and the function.

1. x-axis

2. y-axis

3. linear function

4. hole at $x=a$

5. labels

1. Drawing axes

2. Drawing function line

3. Marking hole

4. Adding labels

1. The function is linear except at the hole

Visual setup of a function with a removable discontinuity to discuss limits.

1. Diagram
   Observation

   Red arrows appear on the x-axis pointing towards 'a' from both sides, and on the y-axis pointing towards 'L'.

1. Red arrows on x-axis

2. Red arrows on y-axis

1. Arrows drawn to show direction of approach

1. The target values 'a' and 'L' remain fixed

Demonstrates the two-sided nature of the limit definition visually.

1. Animation
   Observation

   The speaker erases the initial small graph and draws a larger version of the function curve, axes, and points a and L.

1. function curve

2. x-axis

3. y-axis

4. point a

5. point L

1. The graph is erased and redrawn larger to better illustrate the epsilon and delta distances.

1. The mathematical relationship between x, a, $f(x)$, and L remains the same.

The redrawing is done to provide more space for clearly marking the epsilon and delta intervals on the axes.

1. Animation
   Observation

   The speaker marks a distance $\epsilon$ above and below $L$ on the y-axis and draws horizontal dashed lines.

1. y-axis

2. point L

3. horizontal dashed lines

1. Distances $\epsilon$ are marked above and below $L$, creating an interval $(L - \epsilon, L + \epsilon)$.

1. The function curve and point a remain unchanged.

This visually represents the condition $|f(x) - L| < \epsilon$, meaning the function values must fall within this horizontal band.

1. Animation
   Observation

   The speaker marks a distance $\delta$ to the left and right of $a$ on the x-axis and draws vertical dashed lines.

1. x-axis

2. point a

3. vertical dashed lines

1. Distances $\delta$ are marked around $a$, creating an interval $(a - \delta, a + \delta)$.

1. The function curve, point L, and epsilon markings remain unchanged.

This visually represents the condition $0 < |x - a| < \delta$, defining the neighborhood around $a$ that guarantees the function stays within the epsilon band.

1. Animation
   Observation

   The video shows a pre-drawn graph with axes, a function curve (a line with a hole), and initial labels for L, a, epsilon, and delta. Dashed lines connect these points to illustrate the intervals.

1. Coordinate axes (x and y)

2. Function curve (line with a hole)

3. Points L and a

4. Intervals epsilon and delta

5. Dashed projection lines

1. None; the graph is static initially while the speaker explains the general concept.

1. The relationship shown by the dashed lines: x-values in (a-delta, a+delta) map to y-values in (L-epsilon, L+epsilon).

This visual setup provides the geometric intuition for the epsilon-delta definition before specific numbers are introduced.

1. Animation
   Observation

   The speaker writes specific numbers (2, 1, 0.5, 0.1) onto the existing graph and draws new, tighter dashed lines to represent these specific intervals.

1. New numerical labels: 2 (for L), 1 (for a), 0.5 (for epsilon), 0.1 (for delta)

2. New dashed lines forming a smaller rectangle around the point (1, 2)

1. The abstract intervals are replaced/supplemented with concrete numerical examples.

2. The visual focus shifts to the specific mapping between the interval (0.9, 1.1) and (1.5, 2.5).

1. The underlying function curve remains the same.

2. The logical structure of the epsilon-delta relationship is maintained.

This event grounds the abstract definition in a tangible example, showing exactly how a chosen epsilon leads to a findable delta for a specific function.

1. Animation
   Observation

   The speaker writes the formal limit notation '$\lim_{x \to a} f(x) = L$' on the right side of the screen.

1. Mathematical expression: $\lim_{x \to a} f(x) = L$

1. Transition from graphical/numerical explanation to symbolic representation.

1. The meaning conveyed by the symbol is identical to the previously explained concept.

This connects the intuitive and numerical explanations back to the standard mathematical language used in textbooks.

1. Diagram
   Observation

   A graph of a function with a hole at $x=a$, showing horizontal dashed lines for $L\pm ε$ and vertical dashed lines for $a\pm δ$.

1. Function curve

2. Point (a, L)

3. Horizontal lines $y = L+ε$ and $y = L-ε$

4. Vertical lines $x = a+δ$ and $x = a-δ$

5. Shaded regions for ε and δ ranges

1. Cursor moves along the x-axis within the δ range

2. Cursor moves along the y-axis within the ε range

1. The relationship between the x-range (δ) and y-range (ε) remains constant

The visual demonstrates that keeping x within δ units of a (but not equal to a) forces $f(x)$ to be within ε units of L.

1. Diagram
   Observation

   A coordinate system with yellow axes, a pink/magenta curve, a vertical dashed line at a, a horizontal dashed line at L, a green ε-band, and a purple δ-interval is visible throughout.

2. Formula
   Observation

   Static handwritten formulas remain on screen: $\lim _{x\to a} f(x)=L$, give $ε>0$, They'll give a δ, and 0<|x-a|<δ ⇒ |$f(x)-L$|<ε.

1. Cartesian axes

2. function curve

3. point a on the x-axis

4. point L on the y-axis

5. vertical dashed line through a

6. horizontal dashed line through L

7. ε-band around L

8. δ-interval around a

9. handwritten limit statement

10. handwritten implication

1. No new mathematical writing is added during the clip.

2. The cursor moves among the graph, the δ-interval, the ε-band, and the formulas.

1. The displayed formulas remain unchanged.

2. The geometric roles of a, L, ε, and δ remain fixed throughout.

The visual arrangement links each algebraic symbol to a geometric region: δ controls horizontal proximity to a, ε controls vertical proximity to L, and the implication says the former forces the latter.

1. Animation
   Observation

   The cursor points near the curve and axes early, then moves to the lower-left implication, and later hovers around the ε and δ regions.

2. Audio
   Observation

   The speaker's verbal explanation tracks these pointing motions while discussing closeness of x to a and $f(x)$ to L.

1. Exact cursor path frame-by-frame is approximate because only sampled frames are available.

1. mouse cursor

2. graph region near a

3. ε-band around L

4. δ-interval around a

5. lower-left implication

1. Attention shifts from the overall graph to the formal inequality and back to the ε and δ markings.

1. The underlying board content does not change.

2. The same symbols keep the same meanings while the cursor moves.

The motion serves as a teaching aid, tying the spoken intuition to the written definition without introducing new mathematics.

1. Audio
   Observation

   The speaker explicitly states that the intuitive explanation is 'not rigorous at all'.

Thinking that the visual idea of 'getting closer' is a sufficient mathematical definition for a limit.

A rigorous definition requires precise quantification of 'closeness', which is introduced later in the video series.

1. Audio
   Observation

   The speaker says, 'there doesn't have to be a hole there. The limit could equal actually the value of the function, but the limit's more interesting when the function isn't defined there where the limit is.'

Students might think a limit only exists or is interesting if there is a hole (undefined point) in the graph.

The limit can exist and equal the function's value even if the function is defined at that point. A hole just makes the concept of a limit more distinct from the function's actual value.

1. Audio
   Observation

   Speaker explicitly warns: 'that was just a specific example... But in order for this... by definition... it doesn't just work for one specific instance, it works for any number you give me.'

Believing that finding a delta for one specific epsilon is sufficient to prove a limit.

The definition requires that for *every* positive epsilon, no matter how small, there must exist a corresponding delta. It's a universal quantifier over epsilon.

1. Audio
   Observation

   Speaker explicitly states, 'the one thing I can't guarantee you is what happens when x is equal to a.'

Believing that the limit of a function as x approaches a depends on the value of the function exactly at $x = a$.

The limit only concerns the behavior of the function as x gets arbitrarily close to a, not at a itself. The function might even be undefined at $x = a$.

1. Formula
   Observation

   The displayed condition is 0<|x-a|<δ, not merely |x-a|<δ.

One might think the definition only says x is close to a.

The written hypothesis is the punctured neighborhood 0<|x-a|<δ, so x must be close to a but not equal to a.

1. Formula
   Observation

   The board writes 'give $ε>0$' before 'They'll give a δ'.

2. Audio
   Observation

   The speaker describes first choosing a tiny desired distance and then giving a distance around x where the condition will hold.

One might think δ is chosen first and ε follows.

In the displayed definition, ε is the prescribed tolerance and δ is supplied afterward in dependence on ε.

1. Audio
   Observation

   The speaker transitions from the intuitive explanation to stating the need for a more rigorous definition.

The rigorous epsilon-delta definition generalizes and formalizes the intuitive concept of a limit explained earlier.

1. Audio
   Observation

   Speaker says 'Let me define that with the actual epsilons and deltas and this is what you'll actually see in your math textbook' right before writing the notation.

The verbal and graphical explanation of the epsilon-delta game is the conceptual basis for the formal mathematical notation of a limit.

1. Audio
   Observation

   Speaker introduces the example saying 'just to make this a little bit more concrete'.

The concrete numerical example is a specific application of the general epsilon-delta concept to aid understanding.

1. Formula
   Observation

   Both $\lim _{x\to a} f(x)=L$ and 0<|x-a|<δ ⇒ |$f(x)-L$|<ε are shown together on the same board.

The clip presents the epsilon-delta implication as the formal content behind the limit notation shown above it.

1. Formula
   Observation

   The phrases 'give $ε>0$' and 'They'll give a δ' appear next to the implication.

2. Audio
   Observation

   The speaker explains the process in terms of choosing a tiny distance and then providing a matching distance around x.

The method card describes how the displayed implication is used: prescribe ε first, then find δ so the implication holds.

1. Formula
   Observation

   The implication is written with absolute values |x-a| and |$f(x)-L$|.

2. Audio
   Observation

   The speaker interprets these expressions as distances.

Understanding the implication requires reading the absolute-value expressions as distances on the input and output axes.

1. Audio
   Observation

   Discussion of what the limit means intuitively.

1. Intuitive Definition of a Limit

1. Audio
   Observation

   Speaker mentions the lack of rigor in the previous explanation.

1. Introduction to Rigorous Limit Definition

1. Audio
   Observation

   The speaker defines epsilon as any real number greater than zero representing the distance from L.

1. Epsilon-Delta Definition of a Limit

1. Audio
   Observation

   The speaker defines delta as the distance around a that guarantees $f(x)$ is within epsilon of L.

1. Epsilon-Delta Definition of a Limit

1. Audio
   Observation

   Explanation of epsilon as the desired closeness to the limit L.

1. Conceptual understanding of Epsilon-Delta

1. Audio
   Observation

   Demonstration of finding a delta given an epsilon using a graph.

1. Conceptual understanding of Epsilon-Delta
2. Concrete Numerical Example of Epsilon-Delta

1. Formula
   Observation

   The standard limit notation is presented.

1. Standard Notation for Limits

1. Audio
   Observation

   Warning that the definition must hold for any epsilon, not just one.

1. Misconception: Epsilon-Delta only needs to work for one example

1. Audio
   Observation

   Speaker explains the formal definition using the 'give me epsilon, I'll give you delta' analogy.

1. Epsilon-Delta Definition of a Limit

1. Audio
   Observation

   Speaker clarifies that the limit does not depend on the function's value at $x = a$.

1. Limit depends on value at the point

1. Formula
   Observation

   The board explicitly shows the epsilon-delta implication.

1. Displayed epsilon-delta implication
2. Limit notation

1. Formula
   Observation

   The hypothesis is written as 0<|x-a|<δ.

1. Displayed epsilon-delta implication
2. The condition excludes $x=a$ itself