Intro to Limits One Sided

本站添加: 2026-05-15

[00:00:00]all right hello precalculus we're gonna do something a little different here today just because this is this lesson deals with a lot of graphs and all and I think this might work out better in a different different approach here hopefully you can see this at the top of the page its introduction to limits she'd be so you should print this out and you're gonna follow along as I go through it but before we do this let's just let's just recap and summarize what we've done thus far so this was a lesson on limits okay we're in limits right now and we said there were three types of limits the first type is limits at infinity okay so I know my hand will be in the way sometimes so you know I'll give you an unobstructed view after I write it but limits that infinity was the first type of limit and those limits basically what you're looking what you're doing is you're looking for your horizontal asymptote okay so limits at infinity you're looking for a horizontal asymptote and if there is a horizontal asymptote that's the answer to your limit question if there is no horizontal asymptote like a parabola then the limit does not exist Mean Girls right so the second type of limit there's limits at a specific number right limits at a specific number and in this situation you're not looking for a horizontal ass okay you're not looking for horizontal acetone in this situation you're looking for the Y value that the function is approaching as X approaches that specific number so what do we always try first the first thing that we try is direct substitution direct substitution is what you want to always try first okay the next thing that we try is we try to like factor and simplify because maybe there's maybe there's a hole in the graph or something that that we can kind of simplify out and then do direct substitution after we factor and simplify okay so direct substitution if that doesn't work factor and simplify and then go back to direct substitution okay if there's no way to factor or simplify try rationalizing see if see if you can rationalize the function okay and we did that in algebra two and you can either rationalize the numerator or you could rationalize the denominator okay but that is something we did in algebra two and hopefully we did enough of them in the practice that you're okay with rationalizing and then if you can rationalize and simplify then of course go back to direct substitution okay and then if all else fails you've got to do it with the graph and table okay in your calculator so you put the function in your calculator you go to the table and you approach you approach whatever the x value is from the right hand side and from the left hand side okay I know that looks kind of weird but now we said there are three types of limit problems okay the third type is what we're doing today and these are called one sided limits one-sided limits okay and that's when you approach either just from the right side of an x-value or you approach just from the left side of an x-value okay so let's switch to the lesson here we go alright so at the top you can see it says you've learned about limits at infinity and limits at a specific number next we'll learn one-sided limits and one-sided limits that don't exist okay so here we go one-sided limits Part A says find the limit as X approaches 2 and it says X approaches 2 and there's this little minus sign you probably can't see it in the video but you can certainly see it on your sheet there's a little minus sign as a superscript there okay we're gonna have to figure out what that means and the function that we're we're looking at here is the square root of x minus 2 now that's just simply the square root of x shifted to the right two units so I'm going to graph that right over here okay so one two again we should know certain functions we went through a whole year of learning functions the square root of x looks like this okay and that that that type of thing just knowing about different functions and their graphs that will help you in calculus if you don't you know recognize a graph and you know nothing about it you're gonna struggle in calculus so you want to kind of review some of that you should know that the square root of x looks like that guy right there okay so now we have two questions we have as X approaches two and then there's this little minus sign and in question B it's the limit as X approaches two and there's a little plus sign okay so the question is what the heck does the minus and the plus now we're gonna do we're gonna do Part B before we do Part A so if you give some thought to it these are called one-sided limits so we're gonna have to somehow let you know whether or not we want you to come in from the right side or from the left side of two so guess what that's what the plus and the minus represent the minus and I'll put it right under here over here whenever you see the minus sign okay as a superscript that means from the left okay and whenever you see a plus sign as a superscript that means from the right so like I said we're gonna do Part B first and it says the limit as X approaches 2 from the right okay so if we go to the right of x equals 2 and we get closer and closer and closer what I normally do is I put my pen on the function to the right of 2 and I'm moving to my left because again the plus sign means from the right okay it doesn't mean mean move to your right means from the right so as I approach 2 from the right what Y value is my pen getting closer to and you just kind of just follow it along what Y value am I getting closer to I'm getting closer and closer and closer to a y value of 0 so Part B the answer is 0 okay now let's go back to Part A now this one says what's the limit as X approaches 2 from the left well I can't put my pen on the paper on the function to the left of 2 there is no function to the left of 2 so I can't come in from the left well in that situation that that limit is undefined you can write undefined or does not exist it's up to you undefined or DNA okay take either one of those answers now see this little important note here it says the limit as X approaches any value C does not exist if the limit as X approaches C from the left is different than the limit as X approaches C from the right so if your right-hand limit is not the same as your left-hand limit then the overall limit the limit as X approaches 2 see there's no one-sided limit here this is the overall limit this limit does not exist or you can say undefined but I'm gonna stick with this does not exist thing okay so this overall limit does not exist because the right-hand limit was to 0 and the left-hand limit did not exist since they were not the same then the overall limit does not exist so they have to approach from the left and the right they have to approach the same y-value for the overall limit to exist all right let's do some practice so let's scoop this up a little bit in problems 1 through 4 you want to find F of negative 4 now that's not a calculus problem F of negative 4 is a precal problem that's asking the question when X is equal to negative 4 what is the Y value then we have to do evaluate the left hand limit now the limit is when we approach 4 from the left what Y value are we getting closer to that's different than what Y value exists at x equals negative 4 these are two different questions and then we're gonna do the right hand limit okay so there are three things we're gonna do okay oh I'm sorry and then we're gonna do the overall limit okay so there are 4 things so here we go we've got this graph it says what's F of negative 4 now that's a precalc question now F of negative 4 is what is the Y value when X is equal to negative 4 well you guys tell me what is the Y value when F when X is equal to negative 4 well there's a hole there so that there is actually undefined okay because it's an open circle okay so that's undefined now we get to a limit problem it says what is the limit as X approaches negative 4 from the left so I go to the left of x equals negative 4 and I follow the function along here and as I get closer and closer and closer infinitely close to x equals negative 4 what Y value am i getting closer to so just follow the function and I'm getting closer and closer to positive 4 and then it says what's the limit as X approaches 4 negative 4 from the right so I go on the function and I start approaching negative 4 and we want to get infinitely close to negative 4 what Y value is my pen close to it's also equal to 4 now when the left-hand limit and the right hand limits are the same then the overall limit is 4 ok all right let's take a look at number 2 so the first question is that pre-cal question what is the F of negative 4 well F of negative 4 here is negative 4 okay and it's undefined there because there's an open circle and it's defined down here okay so it's a closed circle so F of negative 4 is negative 3 okay where that closed circle is now let's do the limits all right and and the reason why we're doing this F of negative 4 and then limits these are different questions the limit is all about what are we getting closer to okay so here we go the limit as X approaches 4 from the left so the left-hand limit I get on the function to the left of negative 4 and I approach negative 4 and I'm getting closer and closer and closer and closer to 4 okay so the left-hand limit is 4 and then the right-hand limit well I'm going to get on the function over here and I'm getting closer and closer and closer to negative 4 but what y-value am i getting closer to infinitely close to it's negative 3 now notice the left and the right hand limits are not the same so the overall limit is does not exist okay so pause it kind of digest those first two problems make sure you understand what's going on there let's now move on to 3 ok F of negative 4 again so this is a little tricky but negative 4 is right in between this negative 5 point and that negative 2 point and it's a horizontal line and that horizontal line is at y equals negative 2 so f of negative 4 is negative 2 okay now let's do the left and right hand limits so there is a point right in here here's negative 2 negative 3 negative 4 there's a point right there there's x equals negative 4 that's the only point we're concerned with we don't even care about this stuff of stuff up here or over there so we come down to this graph and we want to come in from the left so since it's it's actually on a horizontal line the Y value is negative 2 and when we come in from the right the Y value is negative 2 it's it's actually not that we're approaching negative to when we come in from the right and from the left we're on negative - okay and therefore the overall limit is negative - okay all right let's look at number four so F of negative four so we have two open circles here here and here and then we have at negative four we have a closed circle and that has a y-value of two so f of negative four is two so we are very concerned about the opening closed circles when we just do F of negative four but when we do limits we really don't care about open and close circles you know you can kind of disregard them so now let's do the limit as X approaches negative four from the left so I'm gonna go to the left of negative four on the function and I'm gonna follow it in and what Y value is my pen getting closer - it's getting closer and closer and closer to four okay now from the right go to the right of negative four I got to be on this piece of the function and I'm getting closer and closer and closer to negative four and my Y value is three now the left-hand limit and the right hand limits they're not equal so this overall limit as X approaches negative four does not exist all right you guys are doing calculus well done let's keep going let's flip it over let's look at five number five says find the limit as X approaches negative three now that doesn't tell us to go from the left to the right that's an overall limit which means we're gonna have to do the left-hand limit as X approaches negative three from the left we have to do that then we have to do the limit as X approaches negative three from the right and then we can answer their question which is the limit as X approaches 3 overall okay so we're gonna have to do three things to answer that question we're gonna go from the left so let's do that if we go from the left as I move towards negative three I'm getting closer to a y-value of two okay from the right as I move closer and closer to x equals negative three I'm getting closer and closer to two so the overall limit as X approaches three overall since the left and the right hand limits are the same it's - all right let's do one more so find the limit as X approaches positive three okay now where the heck is positive three all right so we're gonna have to do the limit as X approaches positive three from the right and we have to do as X approaches positive three from the left I kind of did in a different order there and then the limit as X approaches three overall now this is a little tricky where is x equals three on here it's right on that horizontal line so that was like the one on on the on the front there where the point falls right there in between those two open circles so the left of the we're starting with the right hand limit this time the right hand limit is negative three because it's on that horizontal line it's right in there I don't know if I can there's a closed circle right in there let's put this real close you can see it's right on that on that horizontal line anyway so the right-hand limit is negative three the left-hand limit is also net negative three and therefore overall as X approaches three its negative three okay so let's do a couple of these others down here and I want to leave some for homework okay but here we go let's look at number seven it gives us the graph and it says what is the limit as X approaches three of the line four minus X now when they give you the function and the graph you can do it two different ways this is just a line so therefore can't we do direct substitution of course we can if we put the 3 in for X 4 minus 3 is 1 the answer should be 1 now if you look at the graph it better also be 1 so if you look at the graph as X approaches 3 you have to do the left hand limit I'm gonna put an arrow coming in from the left and a right hand limit coming in from the right what y-value are we getting closer to well if you look across we're getting closer and closer to y equals 1 okay all right now number 8 they don't give us the actual function so the only way we can do this is looking at the graph so we want to do X approaches 2 from the left and from the right so here we go from the left I'm going to just draw an arrow there okay I'm approaching y equals 2 from the right we're approaching y equals 2 so the limit is 2 now if I asked what is f of 2 cuz see this see this closed circle here if I had asked what's F of 2 that's a precalc question that would be 0 because we'd be concerned it's undefined there but it's defined here okay stand by someone is vacuuming and I have to [Music] all right that's the end of the vacuuming all right and next one they give us the actual function x squared plus two that's a parabola and that's a continuous function so direct substitution should work just fine we could just put a 1 in there and we end up with a 3 and now when we look at the graph it better be 3 so let's look so the left-hand limit we're approaching 3 and the right-hand limit we're approaching 3 so overall 3 ok so let's look at number 10 they don't give us the function here just the graph so we're approaching 1 and we want to approach from the left and from the right and what y-value are we getting closer to it doesn't matter that there's an open circle guys this is a limit problem it's about what are we getting closer to we're getting closer and closer to 3 now I I agree if I had asked the question what is f of 1 in this question f of 1 the answer to that question would be 1 but a limit problem is all about getting infinitely close to something ok let's look at number 11 this has an absolute value in there and direct substitution doesn't typically work very well with an absolute value and you can see if you put in a 5 you'll get a 0 in the denominator and that's no good so we go to the graph the left-hand limit is negative 1 the right-hand limit is positive 1 since the left and the right hand limits are different this year does not exist or if you prefer undefined let's look at this one the limit as X approaches 3 of this function direct substitution does not work you would not be able to rationalize this you can't simplify it the only way to do this problem is graphically okay now the left-hand limit wow that thing is going down to infinity okay the right-hand limit that's going up to infinity what y-value is getting closer to it's going up forever this one's going down so the left-hand limit is negative infinity the right-hand limit is positive infinity they're different okay if they were the same you actually can say that the answer is infinity but since it's not a specific number we often just say it does not exist so this one does not exist okay all right moving on as X approaches zero so let's see the left-hand limit is 1 the right-hand limit is 1 so therefore this is 1 I didn't Circle those answers all right that's my compulsive-like compulsiveness I've got to circle all the answers all right let's keep going the limit as X approaches PI over 2 so there's PI over 2 from the left this goes up forever so the Y values are approaching positive infinity from the right it's going down to negative infinity so this is a does not exist the left and right hand limits are not approaching the same number okay even if they were both approaching infinity let's say you can give an answer as infinity even the AP exam accepts that however I usually just write does not exist because what we're looking for is is the Y value a specific Y value that we're approaching right and infinity is not not a specific Y value all right so those were a lot of examples that was good now we have these problems these I'm gonna leave for homework okay now you can do these any way you you know you like but most of them you're going to have to graph I do want you to sketch a graph so like this first function I'll move it down a little bit log X minus 2 now usually you should know what a log graph looks like again you need to review some of this stuff if you forgot well this is just log graph shifted right two units so all log graphs have a vertical asymptote and there's your log graph and there's x equals two okay but what I want you to do is I want you to graph each of these sketch a graph of these some of these I wouldn't expect you to know what this looks like okay or this one or this one down here but I would expect you one over x squared there's the volcano one over X that's the rectangular hyperbola sketch the graph and then answer the limit question okay now don't be fooled down here number 20 that's a limit at infinity you're looking for a horizontal asymptote on that one so all right you get to work on that and well I hope you enjoyed so until next time bye now