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Worked example: Continuity at a point (graphical)

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.A (LO)
,
LIM‑2.A.2 (EK)

Video transcript

we have the graph of y is equal to G of X right over here and what I want to do is I want to check which of these statements are actually true and then check them off and like always I encourage you to pause the video and see if you can work through this on your own so let's look at this first statement so this first statement says both the limit of G of X is X approaches 6 from the right-hand side and the limit as X approaches 6 from the left-hand side of G of X exists all right so let's first think about the limit of G of X as X approaches 6 from the right-hand side as we approach 6 from values greater than 6 so if we look over here we could say ok when X is equal to 9 G of 9 is right over there G of 8 is right over here G of 7 is right over here it looks like it's between negative 3 and negative 4 G of 6.5 looks like it's a little bit it's it's a little bit it's still between negative 3 and negative 4 it's but it's closer to negative 3 G of 6.1 is even closer to negative 3 G of 6.01 is even closer to negative 3 so it looks like the limit from the right hand side does exist so it looks like this one exists now let's see and I'm just looking at it graphically and that's that's all they can expect you to do in an exercise like this now let's think about the limit as X approaches 6 from the left hand side so I could start anywhere but let's say when X is equal to 3 G of 3 is a little more than 1 G of 4 is looks like it's a little bit less than 2 G of 5 looks like it's close to 3 G of 5.5 looks like it's between 5 & 6 g of 5.75 looks like it's approaching 9 and as we get closer and closer as X gets closer and closer to 6 from below from values to the left of 6 it looks like we are we're unbounded we are approaching infinity and so technically we would say this limit does not exist so this one does not exist so I won't check this one off some people will say the limit is approaching infinity but that technically is infinity is not a a value that you can say it is approaching in the classical formal definition of a limit so for this for these purposes we would just say this does not exist now let's see they say the limit as X approaches 6 of G of X exists well the only way that the limit exists is if both the left if both the left the left and the right limits exist and they approach the same thing well we don't even our left limit our limit as export to 6 from the negative side or from the left-hand side I guess I could say it does not even exist so this cannot be true so that's not going to be true the first one's not going to be true G is defined at x equals 6 so at x equals 6 it doesn't look like G is defined looking at this graph I can't tell you what G of 6 should be we have an open circle over here so G of 6 is not equal to negative 3 and this goes up to infinity and we have a vertical asymptote actually drawn right over here at x equals 6 so G is not defined at x equals 6 so I'll rule that one out G is continuous at x equals 6 well you can see that you know it goes up to infinity then it jumps down back down here then continues so just when you just think about it in common-sense language it looks very discontinuous and if you want to think about it more formally in order for something to be continuous the limit needs to exist at that value the vet the function needs to be defined at that value and the value of the function needs to be equal to the value of the limit and neither of these the first two conditions aren't true and so they these can't even equal each other because neither of these exist so this is not continuous at x equals 6 and so the only thing I could check here is none of the above let's do another one of these so the first statement both the right hand and the left hand limit exist as X approaches 3 so let's think about it so x equals 3 is what is where we have this this little discontinuity here this jump discontinuity so let's approach let's go from the positive from values larger three so when X is equal to five G of five is a little bit more negative than negative three G of four is between negative 2 and negative three G of 3.5 is getting a bit closer to negative 2 G of 3.1 it's getting even closer closer to negative 2 G of 3.0 1 is even closer to negative 2 so it looks like this limit right over here now I'm circling the wrong one it looks like this limit exists and in fact it is it looks like it is approaching negative 2 so this right over here is equal to negative 2 the limit of G of X as X approaches 3 from the right-hand side and I'll think about it from the left-hand side so we can start let's I can start here G of 1 looks like it's a little bit greater than negative 1 G of 2 it's less than 1 G of 2.5 is between 1 & 2 G of 2.9 looks like it's it's a little bit less than 2 G of 2.99 is getting even closer to 2 G of 2 point 9 9 9 9 9 9 would be even closer to so it looks like this thing right over here is approaching 2 so both of these limits the limit from the right and the limit from the left exist the limit of G of X as X approaches 3 exists so these are the one-sided limit this is the actual limit now for in order for this to exist both the right and left-handed limits need to exist and they need to approach the same value well this first this first statement we saw that both of these exist but they aren't approaching the same value from the left we are from our sorry from the right we are approaching we approaching negative 2 and from the left we are approaching 2 so this limit does not exist so I will not check that out or I will not check that box G is defined at x equals 3 when x equals 3 we see a solid dot right over there and so it is indeed it is indeed defined there G is continuous at x equals three well in order for G to be continuous at x equals three the limit must exist there it must be defined there and the value of the function there needs to be equal to the value of the limit well the function is defined there but the limit doesn't exist there so it cannot be continuous it cannot be continuous there so I would cross that out and I can't click I can't I wouldn't click none of the above because I've already checked something right or I've actually checked two things already
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