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# Worked example: point where a function is continuous

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

## Video transcript

so we have G of X being defined as the log of 3x when 0 is less than X is less than 3 and 4 minus X times the log of 9 when X is greater than or equal to 3 so based on this definition of G of X we want to find the limit of G of X as X approaches 3 and once again this 3 is right at the interface between these two clauses or these two cases we go to this first case when X is between 0 & 3 when it's greater than 0 and less than 3 and then at 3 we hit this case so in order to find the limit we want to find the limit from the left hand side which will have us dealing with this situation because if we're less than 3 we're in this clause and we also want to find a limit from the right hand side which would put us in this clause right over here and then if both of those limits exist and if they are the same then that is going to be the limit of this so let's do that so let me first go from the left hand side so the limit as X approaches 3 from values less than 3 so I'm going to approach from the left of G of X well this is equivalent to saying this is the limit as X approaches 3 from the negative side when X is when X is less than 3 which is what's happening here we're approaching 3 from the left we're in this Clause right over here so we're going to be operating right over there that is what G of X is when we are less than 3 so log of 3x and since this since this function right over here is defined and continuous over the interval we care about its defining continuous for all X is greater than 0 well we can just substitute 3 in here to see what it would be approaching so this would be this would be equal log of log of 3 times 3 or logarithm of 9 and once again when people just write log here without writing the base it's implied that we're dealing that it is 10 right over here so this is log base 10 it's just a good thing to know that sometimes gets gets missed a little bit all right now let's think about the other case let's think about the situation where we are approaching 3 from the right hand side from values greater than 3 well we are now going to be in this scenario right over there so this is going to be equal to the limit as X approaches 3 from the positive direction from the right hand side of well G of X is in this clause when we are greater than 3 so 4 minus x times log of 9 and this looks like a some type of a logarithm expression at first until you realize that log of 9 is just a constant log base 10 of 9 it's going to be some number close to 1 this is just this expression would actually define a line for X greater than or equal to 3 G of X is just a line even though it looks a little bit complicated and so this is actually defined for all real numbers and so it's and it's also continuous for any X that you put into it so to find this limit to think about what is this expression approaching as we approach 3 from the positive direction well we can just evaluate it at 3 so it's going to be 4 minus 3 times log of 9 well that's just 1 so that's equal to log base 10 of 9 so the limit from the left equals the limit from the right they're both log 9 so the answer here is log log of 9 and we are done
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