Quick question... And this has always been a question I've had with left and right limits. If you're going to plug in the number anyways, why does it matter where the limit approaches from?
It does not really matter in problems like this but I think in problems where you have multiple discontinuities and are required to break it up into 3 or more integrals it will matter. Not sure though.
Sometimes an integral can give you certain functions that are particular when taking limits. For example logs can pop up in the limits and when x would approach zero from both sides. One of the sides diverges to -Infiniti as x approaches zero from the right . But the limit does no exist when approaching the left rendering the entire integral diverging neither to Infiniti or negative Infiniti.
Imagine the function 1/x and you want to evaluate it between [-1, 1]. It is not defined at 0 so you will have to break it up into two integrals. One approaches zero from negative and the other from the positive side. In this case the limit is either negative infinity or positive infinity so it does matter if you come from the left or the right. The same could be happening at a piecewise function in which a certain value is not defined and the function "jumps" between the undefined point thus depending from which direction you approach the limit might be different.
I always thought that if there’s a discontinuity either at the lower or upper limit of integration or between the two, that the integral diverges… I guess this video proves otherwise 😮😮😮
This is beautiful work.
Thank you very much for this detailed explanation, i do understand it ❤️
Quick question... And this has always been a question I've had with left and right limits.
If you're going to plug in the number anyways, why does it matter where the limit approaches from?
It does not really matter in problems like this but I think in problems where you have multiple discontinuities and are required to break it up into 3 or more integrals it will matter. Not sure though.
Sometimes an integral can give you certain functions that are particular when taking limits. For example logs can pop up in the limits and when x would approach zero from both sides. One of the sides diverges to -Infiniti as x approaches zero from the right . But the limit does no exist when approaching the left rendering the entire integral diverging neither to Infiniti or negative Infiniti.
Imagine the function 1/x and you want to evaluate it between [-1, 1]. It is not defined at 0 so you will have to break it up into two integrals. One approaches zero from negative and the other from the positive side. In this case the limit is either negative infinity or positive infinity so it does matter if you come from the left or the right.
The same could be happening at a piecewise function in which a certain value is not defined and the function "jumps" between the undefined point thus depending from which direction you approach the limit might be different.
Great explanation! Thank you!
Thanks so much!! Very helpful
Excellent video - really well explained 😊👍🏽❤️
It helped a lot thanks
nice explanation
Nice
Thanks
thanks so much!!
I always thought that if there’s a discontinuity either at the lower or upper limit of integration or between the two, that the integral diverges… I guess this video proves otherwise 😮😮😮
Sir how are you saying in limit 0 to 1 b approaching to 1 in left hand limit
We come from 0 to 1...and zero is on the left therefore is negative