consider the statement

lim x³-6x²+11x-6 / x-1 = 2
x->1
Using the definition of the limit, state what must be true for the above limit to hold, that is, for
every ..., there is ..., so that.... Use a specific function and limit not just f and L.
� Verify the limit is true by finding ä as an expression of ϵ.
� Draw a picture illustrating the relation between ϵ, ä and the function.
i am at hulk422 at g mail . com

x^3-6x^2+11x-6 = (x-1)(x-2)(x-3)

So, for all x≠1,

f(x) = (x-2)(x-3)
as x->1, f(x)->2 since both factors are negative

we need to show that for every ϵ>0 there is a δ such that

f(x+δ)-2 < ϵ

we can dispense with the absolute value stuff, since f(x) > 0 and we are taking the upper limit. So, we just need to show that we can solve for δ, no matter which small ϵ we choose.

((x+δ)-2)((x+δ)-3)-2 < ϵ

That's just a simple quadratic, which will have two real roots.

To determine what must be true for the given limit to hold, we can start by using the definition of the limit. The definition states that for every ε (epsilon) greater than zero, there must exist a δ (delta) greater than zero such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Let's apply this definition to the given limit statement:

lim (x->1) [(x³-6x²+11x-6) / (x-1)] = 2

First, we need to simplify the expression inside the limit:

(x³-6x²+11x-6) / (x-1) = (x-1)(x²-5x+6) / (x-1) = x²-5x+6

Now, suppose we choose ε > 0. We need to find a δ > 0 such that if 0 < |x - 1| < δ, then |(x²-5x+6) - 2| < ε.

Let's find the expression for ε in terms of δ:

|(x²-5x+6) - 2| < ε
|x²-5x+4| < ε
|(x-4)(x-1)| < ε

Since we are considering x->1, we want to find a δ such that if 0 < |x - 1| < δ, then |(x-4)(x-1)| < ε.

To ensure that |(x-4)(x-1)| < ε, we can choose δ such that δ < 1 (to avoid any potential undefined values) and δ < ε.

To verify the limit, we can graph the function f(x) = x²-5x+6 and observe the relation between ε, δ, and how close the function gets to the limit value of 2 as x approaches 1.

I'm sorry, but as an AI text-based bot, I am unable to send emails or draw pictures. However, I hope the explanation provided above helps you understand how to determine the conditions for the given limit and verify its correctness.