Estimating limit values from graphs

- [Instructor] So we have the graph of Y equals f of x right over here and we want to figure out three different limits and like always pause this video and see if you can figure it out on your own before we do it together. Alright now first let's think about what's the limit of f of x it's x approaches six. So as x, I'm gonna do this in a color you can see, as x approaches six from both sides well as we approach six from the left hand side, from values less than six, it looks like our f of x is approaching one and as we approach x equals six from the right hand side it looks like our f of x is once again approaching one and in order for this limit to exist, we need to be approaching the same value from both the left and the right hand side and so here at least graphically, so you never are sure with a graph but this is a pretty good estimate, it looks like we are approaching one right over there, in a darker color. Now let's do this next one. The limit of f of x is x approaches four so as we approach four from the left hand side what is going on? Well as we approach four from the left hand side it looks like our function, the value of our function it looks like it is approaching three. Remember you can have a limit exist at an x value where the function itself is not defined, the function , if you said after four, it's not defined but it looks like when we approach it from the left when we approach x equals four from the left it looks like f is approaching three and then we approach four from the right, once again, it looks like our function is approaching three so here I would say, at least from what we can tell from the graph it looks like the limit of f of x is x approaches four is three, even though the function itself is not defined yet. Now let's think about the limit as x approaches two. So this is interesting the function is defined there f of two is two, let's see when we approach from the left hand side it looks like our function is approaching the value of two but when we approach from the right hand side, when we approach x equals two from the right hand side, our function is getting closer and closer to five it's not quite getting to five but as we go from you know 2.1 2.01 2.001 it looks like our function the value of our function's getting closer and closer to five and since we are approaching two different values from the left hand side and the right hand side as x approached two from the left hand side and the right hand side we would say that this limit does not exist so does not exist. Which is interesting. In this first case the function is defined at six and the limit is equal to the value of the function at x equals six, here the function was not defined at x equals four, but the limit does exist here the function is defined at f equals, x equals two but the limit does not exist as we approach x equals two let's do another function just to get more cases of looking at graphical limits. So here we have the graph of Y is equal to g of x and once again pause this video and have a go at it and see if you can figure out these limits graphically. So first we have the limit as x approaches five g of x so as we approach five from the left hand side it looks like we are approaching this value let me just draw a straight Line that takes us so it looks like we're approaching this value and as we approach five from the right hand side it also looks like we are approaching that same value. And so this value, just eye balling it off of here looks like it's about .4 so I'll say this limit definitely exists just when looking at a graph it's not that precise so I would say it's approximately 0.4 it might be 0.41 it might be 0.41456789 we don't know exactly just looking at this graph but it looks like a value roughly around there. Now let's think about the limit of g of x as x approaches seven so let's do the same exercise. What happens as we approach from the left from values less than seven 6.9, 6.99, 6.999 well it looks like the value of our function is approaching two, it doesn't matter that the actual function is defined g of seven is five but as we approach from the left, as x goes 6.9, 6.99 and so on, it looks like our value of our function is approaching two, and as we approach x equals seven from the right hand side it seems like the same thing is happening it seems like we are approaching two and so I would say that this is going to be equal to two and so once again, the function is defined there and the limit exists there but the g of seven is different than the value if the limit of g of x as x approaches seven. Now let's do one more. What's the limit as x approaches one. Well we'll do the same thing, from the left hand side, it looks like we're going unbounded as x goes .9, 0.99, 0.999 and 0.9999 it looks like we're just going unbounded towards infinity and as we approach from the right hand side it looks like the same thing is happening we're going unbounded to infinity. So formally, sometimes informally people will say oh it's approaching infinity or something like that but if we wanna be formal about what a limit means in this context because it is unbounded we would say that it does not exist. Does not exist.