Prove the statement using the ε, δdefinition of a limit and illustrate with a diagram like Figur

step 1 Proof at this limit is true using the epsilon delta proof. So how this works is for every epsilo

Prove the statement using the ε, δdefinition of a limit and...

Key Concepts

Algebraic Manipulation in ?-? Proofs

Algebraic manipulation in ?-? proofs involves expressing |f(x) - L| in terms of |x - c| through factorization or other algebraic techniques to determine an appropriate ? in relation to any given ?. This step is critical in demonstrating that the function's values can be made arbitrarily close to the limit L by choosing x sufficiently near c.

Graphical Interpretation of Limits

A graphical interpretation accompanies the rigorous ?-? definition by visually demonstrating how a function behaves near a point. In such diagrams, horizontal bands (of width 2?) and vertical intervals (of length 2?) are drawn to show that as x approaches c, all corresponding function outputs fall within the horizontal band around L. This visual aid reinforces the abstract concept and helps in understanding the proximity conditions required in the ?-? definition.

Limit of a Function

The limit of a function describes the behavior of a function as its input approaches a particular value. It is a foundational concept in calculus that formalizes the idea of approaching a specific output, even if the function is not explicitly defined at that point, ensuring a rigorous approach to understanding continuity and change.

?-? (Epsilon-Delta) Definition of Limits

The ?-? definition is the formal mathematical definition of a limit. According to this definition, a function f(x) approaches a limit L as x approaches some value c if, for every ? > 0, there exists a ? > 0 such that whenever 0 < |x - c| < ?, it follows that |f(x) - L| < ?. This provides a precise way to quantify what it means for a function to approach a limit.

Transcript

00:01 Proof at this limit is true using the epsilon delta proof.

00:04 So how this works is for every epsilon greater than zero.

00:10 So that's the input.

00:11 There needs to be a corresponding delta, the output, such that if zero is less than the absolute value of x minus, in this case free, because it's the limit as x approaches free, is less than delta.

00:32 Then the absolute value of the function, so 1 plus 1 fared x, minus the answer, so minus 2 is less than epsilon.

00:48 So let's start combining like terms, start on the right side.

00:53 So absolute value, so 1 minus 2, so negative 1, so 1 fair to x minus 1 is less than epsilon.

01:02 Now over here on the left, how can i make this look more like the one on the right? well, i can factor out a free.

01:13 Zero is less than three times the absolute value of one ferds x minus one is less than delta.

01:25 And now let's just divide everything by free.

01:28 0 is less than absolute value of 1 fared x minus 1 is less than delta over 3.

01:39 So now you should see at the middle parts are the same in both of them.

01:44 1 fared x minus 1 is less than epsilon and it's less than delta over 3.

01:52 So here we'll have our delta over 3 is equal to epsilon.

02:01 Now we want epsilon to be the input, so let's just multiply both sides by three.

02:08 So delta is equal to free epsilon.

02:13 And we now have, for every epsilon, that's the input, there is now a corresponding delta.

02:20 It's just three times as large.

02:23 And it needs to be positive.

02:25 Both epsilon and delta need to be positive.

02:29 And we have that...

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