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Studying for a test? Prepare with these 4 lessons on Limits basics.
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So we have the graph of a function right over here, and we want to think about what does the limit of f of x as x approaches 3 appear to be? And to do that, let's just think about the limit as x approaches 3 from values less than 3, and from values greater than 3. So let's first think about the limit of f of x as x approaches 3 from values less than 3. So this little negative superscript says we're going to approach 3 from below 3. From 1, 2, 2.5, 2.99, 2.999. So if we approach-- so this is 3 right over here. And we're going take the left handed, or the left sided limit. We're going to approach 3 from this direction first. So when x is 0, f of x is there. When x is 1, f of x is there. When x is 2, f of x is there. When x is 2 and 1/2, f of x is at 5. When x is at-- looks like roughly 2 and 3/4, we get to 4. Looks like about f of x gets to 4.5. And so it looks like as x approaches 3 from values less than 3, it looks like our function is approaching 4. So I would say it looks like the left sided limit of f of x as x approaches 3 is 4. Now let's do the same thing for the right hand side. So the limit of f of x as x approaches 3 from values larger than 3. So notice when x is equal to 5, our f of x is up here. When x is equal to 4, f of x is here. When x is 3 and 1/2, it looks like we're a little under 2 for f of x. And it looks like we're getting closer and closer as x approaches 3 from the positive direction, or from the right side, it looks like f of x is getting closer and closer to 1. So I would estimate, based on this graph, that the limit of f of x as x approaches 3 from the positive direction is equal to 1. Now we have an issue. In order for this limit to exist, we have to get the same value as we approach from the left hand side and the right hand side, but it's clear that we are not approaching the same value when we go from the left hand side as we do when we go from the right hand side. So this limit right over here does not exist. Does not exist. The only way that this would have existed is if we got the same value for both of these, and then the limit would be that value. But we're clearly not getting the same value.