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# Analyzing unbounded limits: mixed function

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.1 (EK)
,
LIM‑2.D.2 (EK)

## Video transcript

so we're told that f of X is equal to x over 1 minus cosine of X minus 2 and there's and they asked us to select the correct description of the one-sided limits of f at x equals 2 and we see that right at x equals 2 if we try to evaluate F of 2 we get 2 over 1 minus cosine of 2 minus 2 which is the same thing as cosine of 0 and cosine of 0 is just 1 and so 1 minus 1 is 0 and so the function is not defined at x equals 2 and that's why it might be interesting to find the limit as X approaches 2 and especially the one-sided limits and if the one-sided limits well well I'll just leave it at that so let's let's try to approach this and there's there's actually a couple of ways you could do it there's one way you could do this without a calculator by just inspecting what's going on here and thinking about the properties of the cosine function and if that inspires you pause the video and work it out and I will do that at the end of this video the other way if you have a calculator is to do it with two is to do it with a little bit of a table like we've done in other example problems so if we think about X approaching 2 from the positive direction well then we can make a little table here where you have X and then you have f of X and so for approaching 2 from values greater than 2 you could have two point one two point zero one now the reason I said calculators these aren't trivial to evaluate because this would be what two point 1 over 1 minus cosine of 2 point 1 minus 2 is 0 point 1 I do not know what cosine of 0 point 1 is without a calculator I do know that cosine of 0 is 1 so this is very very close to 1 without getting to 1 and it's going to be less than 1 cosine is never going to be greater than 1 the cosine function is bounded between negative 1 is less than cosine of X I'll just write the XD I don't need the parenthesis which is less than 1 the cosine of the the cosine function just oscillates between these two values so this this thing is going to be approaching one but it's going to be less than one it definitely cannot be greater than one and that's actually a good hint for how you can just explore the structure here and then you could say all right two point zero one well that's going to be two point zero one over one minus cosine of zero point zero one and this is going to be even closer to one without being one so this could you know this is but it's going to be less than one no matter what cosine of anything is going to be less than it's going to be between negative one and one and it could even be including those things but as we approach as we approach to this thing is going to approach it's going to approach one I guess you could say approach one from below and so you can start to make some intuitions here if it's approaching one from below this thing over here this whole expression is going to be positive and as we approach x equals two well the numerator is positive it's approaching to the denominator is positive so this whole thing has to be approaching a positive value or it could become unbounded in the positive direction as we'll see this is unbounded because this thing is even closer to one than this thing and you would see that if you have a calculator but needless to say this is going to be unbounded in the positive direction so we're going to be going towards positive infinity so these two choices have that and we can make the exact same argument as we go as we approach X in the negative or from below as we approach 2 from below I should say so that's X and that is f of X and once again I don't have a calculator in front of me you could evaluate these things with a calculator it will become very clear that these are positive and as we get closer to two they become even larger and larger positive values and the same thing would happen if you did 1.9 and 1.9 and if you did and if you did 1.99 because here you'd be 1.9 over 1 minus cosine now here you'd have 1.9 minus 2 so this would be negative 0.1 let me scroll over a little bit this second one would be 1 point 9 9 over 1 minus cosine of negative 0.01 0.01 and cosine of negative zero point 1 is the same thing as cosine of 0.1 cosine of negative 0.01 is the same thing as cosine of 0.01 so these two things this is going to be equivalent to that that is going to be equivalent to that it once again we're going to be approaching positive infinity so the only choice where the hell that is true is this first one whether you're approached with your approach 2 from the right-hand side or the left-hand side we're approaching positive infinity now the other way you could have deduced that is say ok as we approach 2 the numerator is going to be positive because 2 is positive and then over here as you approach to cosine of anything can never be greater than 1 it's going to approach 1 would be less than 1 so if this is less than 1 as X approaches 2 it becomes 1 when X is equal to 2 well then this right over here 1 minus something less than 1 is going to be positive so you have a positive divided by positive so you're definitely going to get positive values as you approach 2 and we know or and they've already told us that these are going to be unbounded based on the choices so you would you would also pick that but you should also feel good about it that The Closer that we get to 2 the closer that this value right over here gets to 0 and the closer that this value gets to 0 the closer we get to 1 the closer we get to 1 the smaller the denominator gets and then you divide by smaller and smaller denominators you're going to become unbounded towards infinity which is exactly what we see in that first choice
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