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so we have a function f of X graphed right over here then we have a bunch of statements about the limit of f of X as X approaches different values and what I want to do is figure out which of these statements are true and which of these are false so let's look at this first statement limit of f of X as X approaches 1 from the positive direction is equal to 0 so is this true or false so let's look at it so we're talking about as X approaches 1 from the positive direction so from for values greater than 1 so as X approaches 1 from the positive direction what is f of X well when X is let's say 1 and 1/2 f of X is up here as X gets closer and closer to 1 f of X stays right at 1 so as we pro as X approaches 1 from the positive direction it looks like the limit of X or it looks like the limit of f of X as X approaches 1 from the positive direction isn't a 0 it looks like it is 1 so this is not this is not true this would be true if instead of saying from the positive direction we said from the negative direction from the negative direction the function really does look like it is a the value of the function really does look like it is approaching 0 for approaching 1 from the negative direction when X is right over here this is f of X when X is right over here this is f of X when X is right over here this is f of X and we see that the value of f of X seems to get closer and closer to 0 so this would only be true if we were approaching from the negative direction next question limit of f of X as X approaches 0 from the negative direction is the same as limit of f of X as X approaches 0 from the positive direction is this statement true well let's look our function f of X as we approach 0 from the negative direction let me just in a new color as we approach 0 from the negative direction so right over here this is our value of f of X then as we get closer this is our value of f of X as we get even closer this is our value of f of X so it seems from the negative direction like it is approaching positive 1 from the positive direction when X is greater than 0 oh let's try it out so if like say X is 1/2 this is our f of X if X is let's say 1/4 this is our f of X if X is just barely larger than 0 this is our f of X so it also seems to be approaching f of X f of X is equal to 1 so this looks true they both seem to be approaching the limit of 1 the limit here is 1 so this is absolutely true now let's look at this statement the limit of f of X as X approaches 0 from the negative direction is equal to 1 well we've already thought about that the limit of f of X as X approaches from the negative direction as 0 the limit of f of X as X approaches 0 from the negative direction we see that we're getting closer and closer to 1 as X gets closer and close to 0 f of X gets closer and closer to 1 so this is also true limit of f of X as X approaches 0 exists well it definitely exists we've already established that it's equal to 1 so that's true now the limit of f of X as X approaches 1 exists is that true well we already saw that when we're approaching 1 from the positive direction the limit seems to be approaching 1 we get when x is 1 and 1/2 f of X is 1 when X is a little bit more than 1 it's 1 so it seems like we're getting closer and closer to 1 so let me write that on the limit of f of X as X as X approaches 1 from the positive direction is equal to 1 and now what's the limit the limit of f of X as X approaches 1 from the negative direction well here this is our f of X here this is our f of X it seems like our f of X is getting closer and closer to 0 when we approach 1 from values less than 1 so over here is equals 0 so the limit from the right hand side is a different value than the limit from the left hand side then the villa the limit does not exist so this is this is not this is not true now finally the limit of f of X as X approaches one point five is equal to one so right over here so everything we've been dealing with so far we've always looked at points of discontinuity or points where maybe the function isn't quite defined but here this is kind of a plain vanilla point X is equal to 1.5 that's maybe right over here this is f of 1.5 that right over there is the point well this is the value F of 1.5 we could say F of we could see that F of 1.5 is equal to one that this right here is the point 1 point 5 comma 1 and if we approach it from the from the left hand side from values less than it it's 1 the limit seems to be 1 when we approach from the right hand side the limit seems to be 1 so this is a pretty straightforward thing the graph is continuous right there and so really if we just substitute at that point or we just look at the graph the limit is the value of the function there you don't have to have a function be undefined in order to find a limit there so it is indeed the case that the limit of f of X as X approaches 1 point 5 is equal to 1