Approximating limits using tables

this video we're going to try to get a sense of what the limit as X approaches 3 of X to the 3rd minus 3 x squared over 5 X minus 15 is and when I say get a sense we're gonna do that by seeing what values for this expression we get as X gets closer and closer to 3 now one thing that you might want to try out is well what happens to this expression when X is equal to 3 well then it's going to be 3 to the third power minus 3 times 3 squared over 5 times 3 minus 15 so it x equals 3 this expression is going to be in the C in the numerator e of 27 minus 27 0 over 15 minus 15 over 0 so this expression is actually not defined at x equals 3 we get this indeterminate form we get 0 over 0 but let's see even though the function even though the expression is not defined let's see if we can get a sense of what the limit might be and to do that I'm gonna set up a table so let me set up a table here and actually I want to set up two tables so this is X and this is X to the 3rd minus 3x squared over 5x minus 15 and actually I'm gonna do that again and I'll tell you why in a second so this is gonna be X and this is X to the 3rd minus 3x squared over 5x minus 15 the reason why I set up two tables I didn't have to do two tables I could have done it all in one table but hopefully this will make it a little bit more intuitive what I'm trying to do is on this left table I'm gonna let's try out X values that get closer and closer to 3 from the left from values that are less than 3 so for example we go to 2 point 9 and figure out what the expression equals when X is 2.9 but then we could try to get even a little bit closer than that we could go to 2 point 9 9 and then we could go even closer than that we could go to 2 point 9 9 9 and so one way to think about it here is we try to figure out what this expression equals as we get closer and closer to three we're trying to approximate the limit from the left so limit from the left and why do I say the left well if you think about this on a coordinate plane these are the X values that are to the left of three but we're getting closer and closer and closer we're moving to the right but these are the X values that are on the left side of three they're less than three but we also in order for the limit to exist we have to be approaching the same thing from both sides from both the left and the right so we could also try to approximate the limit from the right and so what values would those be well those would be those would be X values larger than three so we could say three point one but then we might wanna get a little bit closer we could go three point zero one but then we might want to get even closer to three three point zero zero one and every time we get closer and closer so we're gonna get a better approximation for or we're gonna get a better sense of what we are actually approaching so let's get a calculator out and do this and you could keep going two point nine nine nine nine nine nine three point zero zero zero zero zero one now one key idea here to point out before I even calculate what these are going to be sometimes when people say the limit from both sides or the limit from the left or the limit from the right they imagined that the limit from the left is negative values and the limit from the right are positive values but as you can see here the limit from the left are to the left of the x value that you're trying to find the limit at so these aren't negative values these are just approaching the three right over here from values less than three this is approaching the three from values larger than three so now let's fill out this table and I'm speeding up my work so that you don't have to sit through me typing everything into a calculator so based on what we're seeing here I would make the estimate that this looks like it's approaching 1.8 so is this equal to 1.8 as I said in the future we're going to be able to find this out exactly but if you're not sure about this you could try even closer and closer and closer values