Question

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

Answer 1

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.