Continuity 1b - Polynomial/Rational Functions and The Extreme Value Theorem

EDIT: At 4:10, picture is upside down, but idea is same.

Calculus: We note that polynomials and rational functions are continuous where defined. The Extreme Value Theroem and Intermediate Value Theorem are also given with examples.

[MathDoctorBob]

@MultiMoe91 Thanks for the question. Yes. Continuity at a point requires three conditions: 1) the function is defined at the point, 2) the limit exists at the point (and not infinite), and 3) the value of the function at the point is the same as the limit. If we have 0/0, then the function is not defined at the point.

What 0/0 tells us is that there is a chance the limit may exist. If so, we have a removable discontinuity and we can "fix" the hole in the graph. - Bob

[MathDoctorBob]

@MultiMoe91 You're welcome, and thanks for the kind words! Please let me know if you have any other questions. - Bob

[MathDoctorBob]

@MultiMoe91 You're welcome. Glad to be of help.

To see if it removable or not, we have to find the limit at x=4. 0/0 means we need to do more work. The trick here is to multiply by [sqrt(x) + 2] over itself. When you simplify the function becomes (x+1)(sqrt(x) + 2) by canceling the (x-4) terms. Now if we put x=4, the limit becomes 5 x 4 = 20. So this discontinuity is removable by letting F(4) = 20. (Check for the limit: F(4.01) = (.0501)/(.0025) = 20.05.)

[MathDoctorBob]

@MultiMoe91 (cont.) For something not removable, consider F(x) = 1/x at x=0. The limit does not exist at x=0, and no matter how we try to fit a point at x=0, we cannot connect the ends on each side. - Bob

[MultiMoe]

@MathDoctorBob thanks alot Dr bob
that was very help full.
I understand that for the rational function F(x) = (x^2-3^x-4)/ (x)^(1/2)-2
it is continuous everywhere except at the point 4 since substituting 4 in the fuction will give us 0/0.
so the type of the discountinuity is removable?
and i would appreciate it if you tell me how to Redefine the fuction at 4 to make it continuous everywhere :)

[MultiMoe]

what if the rational function is Zero by Zero does that make it a point of discontinuity?

[MathDoctorBob]

@adynizamani You're welcome, and thanks! I've written the opposite of what I said. The IVT applies to both cases, so I'll annotate to clear up. Good eyes. - Bob

[bathingape]

your a pretty intimidating guy for a math teacher....not your typical teacher for sure lol

[adynizamani]

Thanks for the lecture.
I didn't get one thing though. Not so good at maths so I am sorry if it's too basic.

at 4:10, since a>0 then why did you plot "a" on -ve y-axis/+ve x-axis instead of its opposite (since b<0)?

[MultiMoe]

@MathDoctorBob Thank you so much Doctor Bob.
i really appreciate it.
you will always be in my prayers.

[MathDoctorBob]

@bathingape915 The first day of class is always interesting. But I'm pretty much about getting my students to the end of the semester in one piece. - Bob

[adynizamani]

@MathDoctorBob I see, that would be nice. :)