Hello and welcome to the session. In this session we are going to discuss the following question which says that Determine a. Limit sin of x into log of x as x tends to 0. b. Limit log of 1 minus x into cos of pi by 4 x as x tends to 0. So, n-heptance rule states that if the surface and g of x are the functions such that f of a is equal to 0 and g of a is equal to 0 then limit f of x by g of x as x tends to a is equal to limit f dash of x by g dash of x as x tends to a this rule is also applicable for f of a is equal to infinity g of a is equal to infinity With this clear idea we shall proceed with the solution. We are to find the value of the expression limit sin of x into log of x as x tends to 0. Now if we put the value of x as 0 in this expression we get sin of 0 into log of 0 which is of 0 into infinity form since sin of 0 is 0 and log of 0 is minus infinity. Now this can also be written as limit log of x upon 1 upon sin of x to 0. Now again if we put the value of x as 0 in this expression we get log of 0 that is minus infinity upon 1 upon sin of 0 which is 0. So we have 1 upon 0 which is also infinity. So this is of infinity by infinity form. Now according to n-hôpital's rule we have if f of x and g of x are functions such that f of a is equal to infinity and g of a is equal to infinity limit f of x upon g of x as x tends to a is equal to the limit f dash of x upon g dash of x as x tends to a. Now the expression limit x tends to 0 log of x upon 1 upon sin of x is of infinity by infinity form. Therefore applying n-hôpital's rule we have to log log of x with respect to x that is 1 upon x whole upon differential of 1 upon sin of x with respect to x that is minus of cos of x upon sin square of x tends to 0. Now putting the value of x as 0 in the expression we get 1 upon 0 that is infinity whole upon minus of cos of 0 upon sin square of 0 that is 1 upon 0 which is also infinity. So this is of infinity by infinity form. Now this expression can also be written as limit sin square of x upon minus of x into cos of x as x tends to 0. Now if you put the value of x as 0 in the expression we get sin square of 0 which is equal to 0 upon minus of 0 into cos of 0 that is also 0. So this is of 0 by 0 form and according to n-hôpital's rule we know that if f of x and g of x are functions such that f of a is equal to 0 and g of a is equal to 0 then limit f of x upon g of x as x tends to a is equal to the limit f dash of x upon g dash of x as x tends to a. Therefore applying n-hôpital's rule we have limit differential of sin square of x with respect to x that is 2 sin of x into differential of sin of x with respect to x that is cos of x upon differentiating minus of x into cos of x with respect to x using product rule that is we have minus of x into differential of cos of x with respect to x that is minus of sin of x plus cos of x into differential of minus x with respect to x that is minus 1 as x tends to 0 which can be written as limit 2 sin of x into cos of x upon x into sin of x minus of cos of x as x tends to 0 Now putting the value of x as 0 in the expression we get 2 into sin of 0 into cos of 0 whole upon 0 into sin of 0 minus cos of 0 which is equal to 2 into sin of 0 is 0 into cos of 0 which is 1 upon 0 into sin of 0 is 0 minus of cos of 0 which is 1 which is equal to 0 upon minus 1 that is 0 therefore the value of the expression limit sin of x into log of x as x tends to 0 is equal to 0 which is the required answer Now we shall find the value of the expression limit log of 1 minus x into cos of pi by 4x x tends to 0 if we put the value of x as 0 in the expression we get log of 1 minus 0 that is log of 1 which is equal to 0 into cos of 0 which is not defined so this is a 0 into infinity form and this can be written as limit log of 1 minus x into cos of pi by 4x can also be written as 1 upon sin of pi by 4x as x tends to 0 now if we put the value of x as 0 in the expression we get log of 1 minus 0 that is log of 1 which is equal to 0 and sin of 0 is also 0 so this is of 0 by 0 sum now according to L-Hopital's rule if we have f of x and g of x are the two functions such that f of a is 0 and g of a is 0 then limit f of x upon g of x as x tends to a is equal to the limit f dash of x upon g dash of x as x tends to a so applying L-Hopital's rule we have limit differential of the numerator with respect to x that is differential of log of 1 minus x with respect to x we have 1 upon 1 minus x into differential of 1 minus x with respect to x that is minus 1 upon differential of the denominator with respect to x that is differential of sin of pi by 4x with respect to x is cos of pi by 4 of x into differential of pi by 4 of x with respect to x that is pi by 4 as x tends to 0 this can be written as limit minus 1 upon 1 minus x whole upon pi by 4 into cos of pi by 4x as x tends to 0 now putting the value of x of 0 in the expression we get minus 1 upon 1 minus 0 whole upon pi by 4 into cos of 0 which is equal to minus 1 upon pi by 4 into cos of 0 that is 1 which is equal to minus 1 upon pi by 4 that is minus 1 into 4 upon pi which is equal to minus 4 upon pi therefore the value of the expression limit log of 1 minus x into cos of pi by 4x as x tends to 0 is equal to minus 4 upon pi which is the required answer this completes our session hope you enjoyed this session


Hi everyone, Joe for jazby'scasebrakes.com coming at you with the brand new 2022-23 Panini Recon Basketball Six box half case random team break number one a lot of fun stuff here all card ship big Thanks to this group for making it happen Appreciate it all 30 teams are in no combos. Let's roll it randomized names and teams three and a six nine times each one two three four five six No, he does not Jimmy seven eight and nine three and a six nine times for the teams two three four five six seven eight and nine Got rockets down to the jazz after nine All right, Scott with the rocket Zachary with the calves Jonathan with the Nuggets Joe with the Nets and Sons Frank with the Spurs Steven with the Mavericks Matthew with the Sixers Joe with the Pacers Damien with the Timberwolves Gabe with the Raptors Joe with the Wizards Justin with the Kings Steven with the Warriors and Celtics Frank with the Pelicans Joe with the Grizz Simon with the Clippers Scott with the Knicks Fung with the Hornets Robert with the Thunder Frank with the Bucks Joe with the Pistons Las Palmo Joe Damien with the Hawks Kevin with the Bulls Robert with the Magic Robert DeVall is in the break Robert DeVall with the Magic to Eric with my Lakers Justin with the Heat Steven with the Trailblazers and Joe with the Jazz All right, let's order these alphabetically While you're considering trades, we're gonna pop open this fresh case and we're gonna see which six boxes We're gonna do and then we'll pause for a quick trade window The trade window currently open Steelers what's going on on this street? Where's the where's the love for the Seahawks? Clippers up for trade All right, we'll roll the die one two three for the left side four five six for the right side Four one two three four five six, and we'll save these other six for next time All right, we'll pause the video when we come back. We're gonna see if there's any trades Then we'll have the break stick around BRB All right. Welcome back everybody. No deals were done. So the list right here remains the same Thanks everyone for watching for breaking with us for getting into it Appreciate you Let's see what we got in this recon We're looking for the ultra rare hollow dots parallel. Am I gonna know what that's We're gonna know what that looks like as soon as I see it my guess is yes Either way Good luck. Thanks everyone for making it happen Rex was asking earlier any sophomore QB predictions this year refresh my memory who are the sophomore quarterbacks this year We're the quarterback last year I don't know if they were that memorable last year actually Desmond Ritter I feel like ended the season kind of on a high note Like to see him play well. Are they kind of doing a little luminance design right here? Kind of like that Rock the rim Jalen adoring. There's a Keegan Murray to $199 That's right. Can he pick it? Think he showed a lot of promise last year just Kevin Porter junior to $199 and a rookie autograph That's a Blake Wesley for the Spurs 63 out of 99 Future Victor when my name a teammate Frank with the Spurs There's Harrison Barnes to $299 Tyrese Halliburton to $199. I think the sophomore QB Projection for Malik Willis may not be very good unfortunately but he might Get another shot somewhere if they end up trading him there's Dalen Terry to $299 for the Bulls D. Anthony Melton to $299 and your autograph is Cave on Looney 46 out of 49 called it to excellence autograph For the Warriors at Steve Olson The the rebounding machine cave on Looney Wasn't the cave on Looney who was getting like 20 rebounds a night in the playoffs or something like that? Right next box member member Luna Lovegood Harry Potter they called her Looney. It wasn't very nice So they called her kids can be mean Doesn't say one way or the other. I guess there are draft night acetate signature says on the other side of the box They're mostly like this slightly heavier card. So it looks like 55 on the top loaders Marcus smart to $199 It's Clint Capaleta 299 and for the Clippers Musa Diabate 14 out of 25 rookie auto for Simon and the Clippers tried to trade couldn't trade mojo Looks like there can be patches to Scott. All right. Thank you Thank you Sky is the limit for Zion. Well, he just has to stay healthy Come on man. Stay healthy There's Lugans Dort to 49 that's for the Thunder that'll be for Robert Bonzo ball to 299 for the Bulls There's a Chris Middleton To 99 for the Bucks That will be for Milwaukee, that's for Frank Paolo Ben caro 173 out of 299 I like this. This is for Robert and Orlando and the autograph We've got Destin for greatness 82 out of 99. That is Herbert Jones for the Pelicans Frank with the pals. I hope he's destined for greatness Yeah, that he is still I is he even 21 is I'm even 21 so I Think that's why there's still a bit of a Bit of hope 22. Yeah, even though even though the stock is down on Zion I Think is his youth I think kind of gives him Gives people a little There's still a window of hope for him But he's not start turning things around in like the next next season or two RJ Barrett to 299 Is this the this looks like Hollow dots, right? Cam Reddish Allegedly, it's ultra rare. This is for Portland. That's gonna be for Steve Olson Box as it's ultra rare RJ Barrett to 299 and Jabari Smith, Jr. For the Rockets to 299 that is gonna be for Scott True potential autograph to 99 Trevor Kiehl's That's for the Nick Scott Yeah, that's right. There were people that gave up on on Ohtani. We were so I remember You know and myself included because I remember in the shop in our shop. We were selling Ohtani From his like rookie year out of Bowman Baseball or something like that rookie Ohtani's Graded PSA 10s for less than a hundred dollars Probably 50 60 bucks a card and we had a stack of them, too There's a Chet Holmgrim rookie card for okay. See that's for Robert Now Robert got okay. See Anne Orlando magic is the draw. It's pretty good. There's a Walker Kessler to 99 He had a nice season There's Jimmy Harden to 299 and Destin for greatness autograph that is Kenyan Martin, Jr. For the Rockets 24 out of 99. I don't know how far Zion stock has fallen, but I Don't know if it's if it's low enough there's got to be a point where people have to just start buying just to just to speculate just to Just to see what happens another 10 15 minutes to go on this one. We're halfway through The six-box break brand new recon grand of team one We'll have some hoops tomorrow. We'll have another game of the Eastern Conference finals We'll see if the Miami Heat can close things out in Boston Tomorrow night, so we'll be breaking and watching that together. Is this Alperin Sengun to 199 There's Chris Boucher to 299 and for the Timberwolves Matt Ryan 41 out of 99 He had a clutch three for the Lakers early in the season That at the end of this by the end of the season kind of helped them Into the playing where they played the Timberwolves and then ultimately beat them one on nice little nice little run Yeah, I think that was a Someone else did that too Zach Gallin and then I Feel like another pitcher did that this week Maybe for the Guardians or something like that someone else did it Musa Diabate to 99 Bobby porters to 199 and a Franz Wagner 17 I 75 call to excellence autograph for Robert and the Magic Simone Fonte Checo Fonte Chio Fonte Chio 34 out of 299 There's a grounder that hit a bird to not a not a good time for birds Sorry birds Yeah, what are they? I guess those odds shot up. Yeah, well, I think there's been a At least in Southern California. There's been There's been a lot more rain Which has meant a lot more You know a lot more plant growth It's kind of amazing. There's some parts of LA where it's just covered with like wildflowers like in like empty fields and like parks and stuff like that but But more more rain meant more plant life more plant life meant more animals more snakes more rabbits more More birds being born But maybe there's more birds around the country And they're just visiting these big wide open ball parks and they're like it seems like a good place to land and then oops Was it will Brennan that hit a grounder and a bird was just sitting on the field grounds remember last year's playoffs where a The Dodgers in the Padres and a goose giant goose flew and and just landed in the infield But the Anthony Melton and here's Kyrie Irvin 299 Anthony Melton will be for the Sixers. That's from Matthew and behind Dylan Brooks is Is Andrew Nemhard 37 out of 49 that's gonna be for Indiana Joe with the Pacers we named the dog Indiana You were named after the dog I had a lot of fun memories of that dog Paolo Ben Carroll named the dog Indiana Jeremy Granton 99 Walker Kessler to 299 Walker didn't Walker Kessler get some rookie of the year consideration some second third place votes Lonzo And here's one of these autographs that I really like to look of There you go. Moses Moody gold ink autograph nine out of ten for Stephen Olson and the Warriors Warriors come out to play Yeah, yeah, I don't yeah, I think you're right right. I don't think it's not surprising that squirrels end up in in parks ball parks all the snacks and Peanuts and crackerjack laying around There's a good rabbit hole that you can go down well a cat hole I guess that you could go down of cats that end up on Cats that end up on on the field specifically soccer fields The the animals are terrified as if they don't know what they got themselves into but but from a human perspective It's kind of amusing I think there was a cat at a liverpool match many years ago and dubbed the anfield cat. I think was Was ushered off the field. I think I think a nice usher that was on the field kind of scooped him up under his arm the cat Wasn't really fighting and they ended up going to some local Adoption animal adoption place and got adopted pretty quickly that Chet Holmgrim right here This uh, what is this rookie rookie? Looks very psychedelic there Yeah, I find I don't mind the look of this at all either Wasn't there sort of Dave the cat being adopted by John Stones? Doesn't surprise me as the cats end up on the pitch a lot. I feel like that that should be a live stream right the the stadium a ballpark at night and See what kind of animals are wandering around in there out of 49 here some bones Highland That's for Simon in the clippers tried to trade couldn't trade mojo 15 out of 49 There is Santa Clara Jalen Williams for okay see that's for Robert the other Jalen Williams on that team is from Arkansas is Kyrie to 299 I Think Santa Clara Jalen Williams is the one that was drafted higher Kendall Brown to 199 for the Pacers just Kyrie for the math. That's the 299 Devin Vasell Future Victor Wimbanyama teammate. Is that a good thing for these for the guys like Devin Vasell? His rookie year wasn't too long ago Rising tides lift all boats Is Victor Wimbanyama that rising tide could be and a one-of-one True potential one-of-one and Nikola Jovic Not not that not Nikola Jo Kitch This is Nikola Jovic one-of-one Miami Heat Justin with Miami all aboard the big hit Express Nice and what is this world traveler we hachimura Looks like it's a short print isn't it Kevin Durant to 199. There you go. That's a way to make some money You don't think those you don't think those professional sports teams make enough money off of us rex I should do it for free All right, here's a quick little recap nice recon break ladies and gentlemen. Thanks for watching. Thanks for breaking with us I'm Joe. I'll see you next time for the next break. Bye. Bye


on't do well. Now you might be like, well, what about like this one tonight? And it's like, well, this one tonight isn't like per se on sexuality. You know, like, it's not as like directly to that issue. It's more about like, is it okay for you? Yeah. So it's not the same. Come on. And there's certainly not synonymous like this topic. And but Vanessa how Gwynne says, Jeff Durbin for anything to do with pro evangelical. I'd be happy to have Jeff. I think I've gotten a response from him before. I'd be open to hosting him if you wanted to. And so that could be cool. And we'd like to have them on. So that'd be cool. And Scott s. Thanks for coming by says pineapple pizza pineapple. Yes. We could always do that topic. Pineapple on pizza. Maybe. And Andrew says that one comes up all the time. Andrew says I don't really care about gun control, but I have to show. I have a show you can look at to evaluate. That might work Andrew possibly. It depends. It really honestly. And the other thing is we usually have new people. We usually have them in a tag team, at least once before we have a solo debate with them. Just to see like how they'll kind of hold up in the. Kind of spotlight, you could say. And then Anton the wall. Thanks for being with us as well as escape goat. Says what would pull in some subs is James hosting shirtless. Thank you very much for being here. Tina Hodge Stettler. Thanks for being here. Let's see. Jane Joshua Larson says, James going to spice it up with a necrophilia debate. You know that would trigger that would trigger Twitter. And that's why I would enjoy doing it. Let's see.  We have done a debate on incel. It was more of a conversation between a gentleman who was incel and a woman who is an expert on incel stuff. And she was great. She did a fantastic. They both did a fantastic job. I'm trying to remember the reason I bring her up is because she had a popular podcast. I just have to find it. I follow her on Twitter. I just can't remember really where she is. But anime potis on Twitter says, This debate is incredible. I'm so glad they enjoyed it. That's encouraging. But yeah, I should probably go soon. So I do want to say you guys, I want to say thanks for being with me. Thanks for hanging out here. It's always fun to do this after show thing. Steven Seen, I'm going to read your super chat in its fullness. Bubble gaming. Thanks for being with us. We are glad you are here. And let's see here. Good ol' Alex Stein though. That was lively to say the least. And one extra, extra fun one. And yeah, I've got to tell you, that was juicy to say the least. It was really fun and I really do appreciate it. So it was, yeah, I got to tell you, I had a good time. These are fun summer nights when we get to host these big debates like this. This is a wash and Alex is a blast. But let me get back to that super chat. In case Steven Seen, that nasty guy is listening because he wants to hear his nasty comment read. Secular Pagan Mom said, did I miss my comment in super chat? Shout out. I did read yours. No joke. So it is in the Q&A. If you want to go back, you'll be able to find it. And it is in there because I know I for sure remember seeing your name and reading it. And let me go into, let's see, super chat. I'm going to open this up. Nasty guy, Steven Seen. Steen, I actually completely, let's see. He says, he said, appreciate James representing trans beta community. Being born in an alpha's body, but mentally a beta is a tough lot to cast. Stay strong, beta boy. That's a good phrase too. A tough lot to cast. I like that. Is a tough lot to cast. I'm going to use it. I just like figurative language. And it's good job, Steven, in addition to calling me a beta baby or a beta boy. I like that. Is a tough lot to cast. I'm going to use that. And then Samantha said, Alex isn't even synced up. I saw that earlier in the live chat. It's true, Samantha. That was something that I don't know why. It wasn't until after the debate started that I noticed it. And then I was like, ooh, it was like, well, it's too late to restart now and to have Alex leave and come back. So I was kind of like, we're stuck with it. But thanks for your super sticker as well of the rainbow. Appreciate that secular pagan mum. So I do, we appreciate your support. And we do. We want to say, folks, we hope you feel welcome as always. A modern day debate is we really do. We want it to be a place that's fair to our guests as well as we really do want people from different walks of life to be welcomed, whether they be politically left, politically right, Biden backers, Trump supporters, black, white, gay, straight, atheist Christians. You name it. We are glad that you were here. Muslims, it's just, it could go on all night. There's just everybody.  We appreciate it. It makes it, seriously, I'm not joking. It makes it a more fun community here where we have like different perspectives. People are willing to share those. That's a lot of fun. And then I don't know if you guys saw, you guys have probably seen like the Destiny versus Vosh debate that we hosted in person. This was already like two years ago. That was a long time ago. But it was a fun one. And I got to meet both Vosh and Destiny in person. And I really enjoyed it like seriously. Like I had honestly one of the most fun times ever. If you have not seen that, yeah, this is already two years old, which is kind of crazy. I'm like holy smokes. That is so old. But nonetheless, it's an epic little discussion. It was just like the between two ferns kind of like studio. This is, oh, by the way, yeah. And the Wi-Fi didn't work.  Destiny and Vosh in person. The Wi-Fi didn't work. And we're fortunate that Destiny saved the day. He let us use his data to livestream the entire debate. So that was like seriously. We're really grateful for that in person debate. If you guys haven't seen that, have you really not seen that for real? It's out there. But that was one of my favorite in-person debates. So I want to say thanks guys. I love you guys. Thanks for all your support. I got to go. It's getting late here. It's 9.24 where I am in mountain time. So I love you guys. I appreciate you. Thanks for all of your guys' support of me personally and the channel. You guys make this fun and seriously, it's a blast. We're excited. You guys, we have a vision of providing a neutral platform so that everybody has their chance to make their case on a level playing field. Especially because folks, it's think about it. Mainstream media, I think that we at Modern Day Debate can provide a more neutral kind of platform where we give everybody a fair shot. We try to be nonpartisan. And that's because you guys deserve a better class of debate channel or coverage loosely speaking of the news. And you might say, James, come on. How can you even compare yourself to CNN or Fox News or whatever it is? Well, hey, you know, it took Joe Rogan about 13 years to get where he is. And we are only three years into this. So we are just getting started, my dear friends. This is only the beginning of our story. We are absolutely thrilled about the future. We are continuously working on improving things. That's important to us because it's true. There are things that we can improve on, and we will as we look more carefully at how we do them. As I mentioned, we're working on putting together some in-person debates or kind of panels that are going to be epic. The production value is going to be amazing. It's going to be our version of middle ground by Jubilee, if you've probably seen their videos. They're going to be amazing. Want to say, though, I love you guys. Seriously, we're excited about the future. We've got a lot of cool stuff coming up. Thank you guys for all of your support. I'm excited to see you the next time we have a debate, which we have for sure won this Saturday. That's our plan. That's pictured at the bottom right of the screen. We might also have one on immigration this Friday night. So you don't want to miss them. If you haven't yet, hit that subscribe button. Thanks, guys. I love you and excited to see you for the next one.