I am working through a khan academy calculus course and I am a but stumped at one of their videos. I linked it below. It seems that they are using th terms "from above" and and "from below" interchangably. If not, what is that allows someone to determine the limit of a nested function when the two conditions of the theorem are not met (condition 1: there is no limit in the nexted function condition 2: the function does not continue past the stated x-value for which we are determining the limit)?


Thank you for your help in advance!



 Nov 30, 2020

This video seems to make it more difficult than necessary but it is probably doing that on purpose.

If you get your head around difficult terminology it usually means you are learning properly, hopefully at a deeper level.



A better graph representation:



\(\displaystyle \lim _{x\rightarrow 0}\;f(g(x))\\ \text{I would just have said that } g(0)=2 \;\;so\;\;\\ =\displaystyle \lim _{x\rightarrow 0}\;f(2)\\ =0\)


Let's see what the video teacher was saying


\(\displaystyle \lim _{x\rightarrow 0}\;f(g(x))\\ \)

Look at the g graph first.

I am just referring to red stuff that he talked about.  He has not explained it well.


look at g function first     (vertical axis is g, horizontal axis is x)

As x approaches 0 from the negative x side (he has called the left side the below side.)

g approaches 2 from higher g values,... from above   



Now move over to the f function

If you are going to find f(g(x)) then the horizontal axis become   g(x) or g      NOT x  and the vertical axis is f(g(x)) or just f for short.


So g is approaching 2 from above  means that on the f graph, g is approaching 2 from the right side.  


See if you can get your head around that.






\displaystyle \lim _{x\rightarrow 0}\;f(g(x))\\
\text{I would just have said that } g(0)=2 \;\;so\;\;\\
=\displaystyle \lim _{x\rightarrow 0}\;f(2)\\

 Dec 1, 2020
edited by Melody  Dec 1, 2020
edited by Melody  Dec 2, 2020

Thank you Melody! It doesn't make sense YET, but I will re-read your explanation a few more times and I'm sure that it will!

WillyGolden  Dec 1, 2020

I've added an amended graph that might help you a bit.

Melody  Dec 2, 2020

Thanks, Melody.....that was a LOT of work   !!!!


cool cool cool

CPhill  Dec 2, 2020

Thanks Chris,      blush

It wasn't that much really.  Thinking it through required a bit of effort though.

I hope Willy, or any other interested person, can work through what I have said.    laugh

Melody  Dec 2, 2020

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