The discussion centers on determining the largest number that can fill the blank in the statement "If x is within _______ units of 3 (but not equal to 3), then f(x) is within 0.01 unit of 2." The function provided is F(x) = (x^3 - 7x^2 + 17x - 15)/(x - 3), which has a removable discontinuity at x = 3. The conclusion is that the largest number that can fill the blank is 0.01, allowing f(x) to remain within the specified range of 2 when x approaches 3.
PREREQUISITESStudents studying calculus, particularly those focusing on limits and continuity, as well as educators looking for examples of removable discontinuities in functions.
This question doesn't make sense unless you are given a specific function f! It looks to me like f is some continuous function such that f(3)= 2. But what goes in the blank depends upon exactly what f is. Surely you realized that?MellowOne said:Homework Statement
For the statement "If x is within _______ units of 3 (but not equal to 3), then f(x) is within 0.01 unit of 2," write the largest number that can go in the blank
Homework Equations
Nonr
The Attempt at a Solution
It's a multi-step problem, but the two numbers I ended up with are 1.99 or 2.01. I'm not sure which one fits into this statement because I'm not quite sure what it's asking.
The exclamation point after f was punctuation in the sentence, just like this one!MellowOne said:Sorry, maybe I should have put the function that I was given. I'm not sure what F! means, but here's the function I'm given. F(x) = (x^3 - 7x^2 + 17x - 15)/( x - 3) It is a continuous graph, but at x = 3 there's a point of discontinuity which is (3,2).
You mean that F has a removable discontinuity at x= 3. That's a discontinuity because 33- 7(32+ 17(3)- 15= 27-63+ 51- 15= 78- 78= 0 as well as 3- 3= 0: both numerator and denominator are 0 at x= 3. Because x= 3 makes the numerator 0, we know that x-3 is a factor. Knowing that it is easy to see that x3- 7x2+ 17x- 15= (x-3)(x2- 4x+ 5) so for x NOT equal to 3, this is just F(x)= x2- 4x+ 5= (x-4)(x-1). The discontinuity is "removable" because that has limit 3 at x= 2.MellowOne said:Sorry, maybe I should have put the function that I was given. I'm not sure what F! means, but here's the function I'm given. F(x) = (x^3 - 7x^2 + 17x - 15)/( x - 3) It is a continuous graph, but at x = 3 there's a point of discontinuity which is (3,2).