In general, the limit of a function at 'a' has nothing to do with its value at 'a', unless the function is continuous at 'a', which happens when the value at 'a' coincides with its limit at 'a'. In the limit, you only care about values AROUND 'a', different than 'a'. The limit may NOT exist at 'a', even if f(a) is defined and the limit at 'a' may exist even if f(a) is not defined.

Hi Phil! It's great to have you in the study group. I'm glad you enjoyed some of the earlier ones that Arben taught. What you are working on with impedance sounds very interesting. I hope you enjoy the study group.

Carl, if you go to the online "Introduction to Calculus" course, select , select Lesson #4, and jump to 8:25 in the video, you'll see the Devendra discussing one-sided limits. At the end of that section @9:06, Devendra notes,

Because you don't have the same behavior on the left and right, you get back indeterminate as the answer for the limit.

5-4-23
Dear Professor Kapadia,
Great course. I'm a newbie and learning. About Riemann integrability, I read that Lebesgue integrability is also a possibility if the Lebesgue measure is 0.
This then allows Riemann integrability and this justifies a closed interval, but with a Weierstrass covering. The points at which there is no differentiability are allowed f they have Lebesgue measure 0. But can the genus be above 0 ever.
Apologies if my terminology is a bit off.
My basic question is whether Lesbesgue integrability is of use beyond Riemann integrals?
Thanks,
Paul Shapshak, PhD
USF

But the function is defined at x=-3 as being equal to 6/(x+4). Therefore it's limit at x=-3 is given by that equation and should be equal to 6. In fact if you evaluate the function at x=-3 that's the answer you get.

You need to include the assumptions, otherwise Mathematica will treat n like a generic variable. For example, consider the following:

In[1]:= Limit[1/x, x -> a]
Out[1]= 1/a

That result is true for all values of a except for 0.

In[4]:= Limit[1/x, x -> 0]
Out[4]= Indeterminate

I took Quiz 5 today; the last issue I had was understanding solving Mean Value Theorem problems in Mathematica. I generally use the transcripts of the video sessions to scrutinize the WL code. The example in Lesson 17 in the "Application" section of the transcript shows the following WL code for the example:

f[x_]:=x^3-8x

sol = Solve[f[4]-f[0] == f'[x](4-0) && 0<=x<=4,x]

I think this Solve statement is structured in a terrible way. It took me a long time to realize that the "(4-0)" was part of the slope calculation. I found a superior explanation also authored by @Devendra Kapadia : https://www.wolfram.com/broadcast/video.php?c=105&p=6&v=3188 . It used the following WL code fragment -- from ~ 1 minute into the video:

f[x] := x^3 - 3 x

{a, b} {-2, 2};

sol = Solve [ f[b]-f[a] / b - a == f'[c] && a < c < b, c]

The Solve statement is laid out even prettier in the video: the slope calculation is shown with a 2-line fraction with the numerator directly over the denominator. Aha! That way of expressing the code is lucid. It makes a huge difference to show the code this way. I also liked how he used a, b, and c: it was much easier to see that a and b were the secant endpoints and c was the x value at the tangent point -- far better than slinging around constants for the coordinates in the Solve statement.

I just wanted to grumble. In the best of all worlds, that bad example would be edited out of the course and replaced with a lucid one. I think I had an expectation that the code would always be perfect in the examples; I'll be a bit more skeptical in the future. The good news: I don't think I'll ever forget these particular calculus ideas and exemplary WL code fragment now. :)

One other note for @Luke Titus: I noticed that a student in the course asked a question in a new discussion with the title "About a list of Calculus Functions." I thought you might have missed that question. FYI.

Luke answered this during the chat session during the course today: Format->ScreenStyleEnvironment->"Working" will set the scratch notebook so that scrolling works. Note to the @Wolfram U team: I believe it would be helpful to have this be the default mode for any scratch notebook provided for Wolfram's online courses. I also recommend setting the screen style environment to "Working" for all notebooks published for courses. It took me a couple of months when starting out to realize that the scrolling style was settable in Mathematica.